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How Long Is It? Difficulties with Conventional Ruler Use in Children Aged 5 to 8

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Title: How Long Is It? Difficulties with Conventional Ruler Use in Children Aged 5 to 8
Language: English
Authors: Gómezescobar, Ariadna (ORCID 0000-0001-5104-6269), Guerrero, Silvia (ORCID 0000-0003-1569-1089), Fernández-Cézar, Raquel (ORCID 0000-0002-9013-7734)
Source: Early Childhood Education Journal. Nov 2020 48(6):693-701.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 9
Publication Date: 2020
Document Type: Journal Articles
Reports - Research
Education Level: Early Childhood Education
Elementary Education
Kindergarten
Primary Education
Descriptors: Difficulty Level, Young Children, Instructional Materials, Measurement, Kindergarten, Elementary School Students, Instructional Program Divisions, Mathematics Skills
DOI: 10.1007/s10643-020-01030-y
ISSN: 1082-3301
Abstract: The ruler is the standard measuring instrument used for measuring lengths. However, measuring lengths with a ruler is a challenge for children. For this reason, this study explores how children who have not received specific school instruction on its use, measure lengths with a conventional ruler. The relative object-ruler position, the strategy to justify the measurement, and the combination between them are analysed. Additionally, the possible influence of the grade and the transition from Kindergarten to Primary School is also studied. To achieve this aim, 99 children were asked to measure a cardboard strip in both a free and a directed situation. The results showed that in free measurements children tend to situate the object in the 2 hash mark of the ruler, the reading of the endpoint was identified as the most used strategy, and the combination of this strategy with lining up the object at 0 was the most commonly used in correct measurements. On the other hand, the results also showed marginal significant differences between age groups in such a way that children in the last year of kindergarten measured better than those in the first year of primary school. To conclude, the educational implications of these results are discussed.
Abstractor: As Provided
Entry Date: 2020
Accession Number: EJ1273848
Database: ERIC
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  Value: <anid>AN0146531279;5mx01nov.20;2020Oct22.05:06;v2.2.500</anid> <title id="AN0146531279-1">How Long Is It? Difficulties with Conventional Ruler Use in Children Aged 5 to 8 </title> <p>The ruler is the standard measuring instrument used for measuring lengths. However, measuring lengths with a ruler is a challenge for children. For this reason, this study explores how children who have not received specific school instruction on its use, measure lengths with a conventional ruler. The relative object-ruler position, the strategy to justify the measurement, and the combination between them are analysed. Additionally, the possible influence of the grade and the transition from Kindergarten to Primary School is also studied. To achieve this aim, 99 children were asked to measure a cardboard strip in both a free and a directed situation. The results showed that in free measurements children tend to situate the object in the 2 hash mark of the ruler, the reading of the endpoint was identified as the most used strategy, and the combination of this strategy with lining up the object at 0 was the most commonly used in correct measurements. On the other hand, the results also showed marginal significant differences between age groups in such a way that children in the last year of kindergarten measured better than those in the first year of primary school. To conclude, the educational implications of these results are discussed.</p> <p>Keywords: Conventional ruler; Length measurement; Kindergarten education; Primary school; Measurement strategies</p> <hd id="AN0146531279-2">Introduction</hd> <p>Measuring lengths is a common task in daily life (Northcote and Marshall [<reflink idref="bib26" id="ref1">26</reflink>]). The importance of measurements in mathematical education is stated in different documents and laws (LOMCE [<reflink idref="bib23" id="ref2">23</reflink>]; NCTM [<reflink idref="bib25" id="ref3">25</reflink>]), but despite the recognised conceptual difficulty that length measurements entail for children (Bragg and Outhred [<reflink idref="bib3" id="ref4">3</reflink>], [<reflink idref="bib4" id="ref5">4</reflink>], [<reflink idref="bib5" id="ref6">5</reflink>]; Chamorro and Belmonte [<reflink idref="bib7" id="ref7">7</reflink>]; Cullen and Barrett [<reflink idref="bib10" id="ref8">10</reflink>]; Gómezescobar et al. [<reflink idref="bib13" id="ref9">13</reflink>]; Ho and Lowrie [<reflink idref="bib15" id="ref10">15</reflink>]; Kamii [<reflink idref="bib17" id="ref11">17</reflink>]; Kotsopoulos et al. [<reflink idref="bib20" id="ref12">20</reflink>]; Levine et al. [<reflink idref="bib22" id="ref13">22</reflink>]; Sisman and Aksu [<reflink idref="bib29" id="ref14">29</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref15">30</reflink>]), both teaching and research are not as robust in the field of measurement as in that of arithmetic (Barrett et al. [<reflink idref="bib1" id="ref16">1</reflink>]). This is why this paper aims to increase knowledge about children's notions of measurement.</p> <p>Length is a one-dimensional property that limits the space between the two ends of an object. Like all magnitudes, it can be compared and quantified. For Piaget et al. ([<reflink idref="bib28" id="ref17">28</reflink>]), the notions of conservation and transitivity are key issues for the understanding of length measurement. Subsequently, Lehrer ([<reflink idref="bib21" id="ref18">21</reflink>]) removes the focus from the measurement process and focuses on the units, putting forward eight ideas needed to understand the measurement. These ideas are not hierarchical, as the author considers them evenly important: unit-attribute relations, iteration, tiling, identical units, standardisation, proportionality, additivity and origin or zero point. This last idea is related to the fact that the origin should not influence the measurement; that is, any point can be the origin of the measurement. The measurement origin is especially important because it is a challenge for children when they measure with rulers (Bragg and Outhred [<reflink idref="bib4" id="ref19">4</reflink>]; Kamii [<reflink idref="bib17" id="ref20">17</reflink>]). Likewise, in measurements with a ruler, it is also crucial to consider another important concept: the accumulation of distance. Stephan and Clements ([<reflink idref="bib31" id="ref21">31</reflink>]) define distance accumulation as the number of times the unit has been iterated from the starting point of the measurement to the endpoint, so this concept is closely related to the origin or zero point.</p> <p>The ruler is an instrument that is built by the iteration of a unit, usually centimetres or inches, without overlapping or gaps, which is quantified by numbering. Therefore, the ruler could be considered a ready-made instrument. However, as previously stated, children make mistakes in their use. In these studies, there are also presented low percentages of kindergarten students who measure with a ruler correctly. Besides, Boulton-Lewis et al. ([<reflink idref="bib2" id="ref22">2</reflink>]) pointed to the children's preference for the use of standard units even though teachers emphasised the use of non-standard units. For these reasons, we plan to study the students' interpretation of the ruler prior to instruction, in order to be able to later adapt the instruction to the children's knowledge.</p> <p>In this paper, which is part of a broader study on the measurement of length in early ages, we study the measurements made with a conventional ruler by children between 5 and 8 years of age. In the case of Spain, this age range is particularly attractive for two reasons. The first reason is that these students are younger than those for which the curriculum expressly requires knowing how to measure correctly with a conventional ruler (LOMCE [<reflink idref="bib23" id="ref23">23</reflink>]). In this context, it is required in 2nd grade (Decree 54/2014, [<reflink idref="bib12" id="ref24">12</reflink>]), but in stages prior to this, the curricula encourages students to measure with non-standard units rather than with standard ones. The second reason is that, in this time period, children experience the transition between kindergarten, from 3 to 6 years old, to primary school, from 6 to 12 years old. This change of stage generally entails recognised differences in teaching practices (González et al. [<reflink idref="bib14" id="ref25">14</reflink>]; Tamayo [<reflink idref="bib32" id="ref26">32</reflink>]) that could affect, even indirectly, their interpretation of the ruler. These two factors have been taken into consideration for the selection of the participants' ages in the current study.</p> <p>The quantitative length-measurement studies found in the literature involve generally artificial tasks, which are mainly paper-based (Bragg and Outhred [<reflink idref="bib3" id="ref27">3</reflink>], [<reflink idref="bib4" id="ref28">4</reflink>]; Cullen and Barrett [<reflink idref="bib10" id="ref29">10</reflink>]; Ho and Lowrie [<reflink idref="bib15" id="ref30">15</reflink>]; Irwin et al. [<reflink idref="bib16" id="ref31">16</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref32">30</reflink>]). This fact does not allow the child to experiment with the material and play with his or her notions on the evaluated topic. Furthermore, when the studies use manipulatives, they are often altered, as in the case of the broken ruler (a ruler with the zero missing) used in several studies (Bragg and Outhred [<reflink idref="bib3" id="ref33">3</reflink>], [<reflink idref="bib4" id="ref34">4</reflink>]; Irwin et al. [<reflink idref="bib16" id="ref35">16</reflink>]; Nunes et al. [<reflink idref="bib27" id="ref36">27</reflink>]). The broken ruler forces the starting point not to be zero. In order to approach a measurement situation as real as possible, where children can show their spontaneous knowledge of measurement, in this study, we used no altered manipulative material, and instead used a conventional ruler for use in a free situation and a directed one.</p> <p>Exploring children' errors when they are measuring with a ruler is one of the most interesting aspects of analysing children's knowledge of that instrument during their school years. The main difficulties encountered in the literature are related to the conceptualisation of the unit and the origin of the measurements (Bragg and Outhred [<reflink idref="bib5" id="ref37">5</reflink>]; Cogndon et al. [<reflink idref="bib9" id="ref38">9</reflink>]; Levine et al. [<reflink idref="bib22" id="ref39">22</reflink>]; Nunes et al. [<reflink idref="bib27" id="ref40">27</reflink>]; Sisman and Aksu 2015; Solomon et al. [<reflink idref="bib30" id="ref41">30</reflink>]). This information can be obtained through the analysis of the strategies used by children in their measurements. The strategies that address notions about length measurement most often referred to in the reviewed studies (Cogndom et al. [<reflink idref="bib9" id="ref42">9</reflink>]; Cullen [<reflink idref="bib11" id="ref43">11</reflink>]; Cullen and Barrett [<reflink idref="bib10" id="ref44">10</reflink>]; Irwin et al. [<reflink idref="bib16" id="ref45">16</reflink>]; Kamii [<reflink idref="bib17" id="ref46">17</reflink>]; Levine et al. [<reflink idref="bib22" id="ref47">22</reflink>]; Sisman and Aksu [<reflink idref="bib29" id="ref48">29</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref49">30</reflink>]) are: reading of the number matching the right end of the object or reading of the endpoint, counting all the marks along the object, and the action on the interval, e.g. pointing or sweeping the finger along the interval. The first strategy, reading of the end point, is closely related to the measuring procedure adults generally use, that is, placing one end of the object at the zero mark and reading the number matching the other end. Therefore, the correction of the measurement using this strategy depends on where the child places the object. Kamii ([<reflink idref="bib17" id="ref50">17</reflink>]) found that more than half of the 4th graders used the physical edge of the ruler as the starting point for their measurements. However, the tendency for the Bragg and Outhred ([<reflink idref="bib4" id="ref51">4</reflink>]) 6 to 8 old-year children was to place the object on the <emph>1 mark</emph>. Therefore, the children who exhibited difficulty with the initial interval did not pay attention to the object and ruler relative position when using the measuring strategy, as was evidenced in their enunciation of the result. That is, children show they undergo this difficulty when they say that an 8 cm object measures 10 cm, as a result of the fact that the object's left edge is placed on the 2 mark, instead of saying that it measures 8 cm. The second strategy, counting marks, could exhibit a difficulty when they say the 8 cm object measures 9. The reason could be that they are perceiving the marks as the unit of measurement, and not the space interval between two marks. Therefore, it could entail a lack of unit perception. Children believing that marks are the unit of measure has been reported by Bragg and Outhred ([<reflink idref="bib5" id="ref52">5</reflink>]), who asked 6th graders to point out the centimetres of a ruler, and the majority, 68%, indicated the marks. The last strategy, action on the interval, reveals that the child is identifying the interval as the unit, so this strategy could be considered as the one with the biggest conceptual load in terms of length measurement. However, most research on children's measurement strategies infers the strategy from the child's numerical answer (Cogndom et al. [<reflink idref="bib9" id="ref53">9</reflink>]; Irwin et al. [<reflink idref="bib16" id="ref54">16</reflink>]; Kamii [<reflink idref="bib17" id="ref55">17</reflink>]; Levine et al. [<reflink idref="bib22" id="ref56">22</reflink>]; Sisman and Aksu [<reflink idref="bib29" id="ref57">29</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref58">30</reflink>]), without asking students directly how they reached their measurement. To avoid the researchers' inferences and to get more spontaneous information, in this study, the children were explicitly asked about the strategy they followed in their measurements, since the verbalisation of their actions provides specific information about their conceptualisation.</p> <p>The general purpose of this work is to test how 5- to 8-year-old children, without previous school instruction, navigate real situations of measurement with the conventional ruler, focusing on the analysis on the object-ruler relative position and the strategies children use to measure. Evidence will be obtained through the communication of the results. The other focus will be the combination between relative position and the strategy that children apply when measuring correctly. The possible influence of the grade or age group on the measurements and the transition from kindergarten to primary school is also studied.</p> <hd id="AN0146531279-3">Materials and Methods</hd> <p>The methodology used is quantitative, based on absolute and relative frequencies, as well as on percentages. The possible relationship between variables and factors are analysed by correlation coefficients.</p> <hd id="AN0146531279-4">Sample</hd> <p>Convenience sampling was used to carry out this study, and the sample was drawn from students in the public school system, as it is the most common in Spain and because of the availability of teachers and families of children willing to participate in the study therein. For this study, one school within the province of Toledo (Castilla-La Mancha -Spain-) and another school located in the capital city of Toledo participated. The initial sample consisted of 102 children between the ages of 5 and 8. As three of the children did not completely respond to the interview, they were eliminated from the analyses. Therefore, the final sample consisted of 99 students (48 girls) enrolled in three different grades: 3rd Kindergarten, (5–6 years old; N = 37; EM = 67 months), 1st Primary School (6–7 years old; N = 28; EM = 78 months) and 2nd Primary School (7–8 years old; N = 34; EM = 91 months). The children were part of a broader transnational study on length measurement in early ages with students from Spain and Portugal. In this paper, the data related to Spanish children are presented.</p> <p>The participants, as their teachers orally confirmed, had not received any school instruction on the use of the ruler in their school trajectory. That is, no group had ever performed a length measurement by using a ruler inside the school. Nonetheless, the possibility that children would have used the ruler at home was not controlled. However, the children were familiar with the instrument since the ruler was an object present in the classroom and used on occasion, for example, to draw straight lines.</p> <hd id="AN0146531279-5">Procedure</hd> <p>Each child was interviewed and videotaped individually. The children took two measurements. For the first task, called Free Measurement (FM), they were given a strip of cardboard 8 cm long to be measured with a 30 cm long conventional ruler situated on a table. They were asked to freely measure the cardboard strip to answer the question: "How long is it?". Then, they were asked for their justification," How did you know?" For the second task, called Measurement from Zero (ZM), no matter what the participant previously did, the interviewer lined up the cardboard strip to the zero point on the ruler and asked the child: "How long is it? Then she asked again: "How did you know?" The interview took around 45 s for the kindergarteners, around 40 s for the 1st graders and around 35 s for the 2nd graders.</p> <p>The answer to the second question was considered the verbalisation of the thinking strategy employed by the child. These strategies were categorised according to Cullen and Barrett ([<reflink idref="bib10" id="ref59">10</reflink>]), as reading of the endpoint (EP), when the child indicated the number on the ruler closest to the final point of the object, pointing it out or reading it, or read the complete numerical sequence contained between the two ends of the object; as counting marks (CM), when the child counted the marks on the ruler spanning the length of the object, by pointing them out or counting out loud and as Other (OTH), when the child gave non-mathematical arguments (e.g., "measures 5 because I am 5-years-old"), did not answer or answered "I do not know".</p> <p>The position at which the child aligned the left end of the object on the ruler was analysed only in the Free Measurement. As a result, four categories were created: -1, when end of the object was placed matching or approximately matching the physical end of the ruler; 0, when it was located at zero or close to it, either to the right or left of 0; 1, when it was located at 1 or close to it and 2, when it was located near 2 or at any other number further than 2.</p> <hd id="AN0146531279-6">Statistical analysis</hd> <p>The data obtained were analysed with Statistical Package for Social Sciences, SPSS, v. 24. The number of correct answers was considered a numerical variable, which was counted as a percentage. The comparisons between Measurement from Zero and Free Measurement were performed by using tests for related samples, since these measurements were consecutive. The possible association between the number of correct answers and the factors studied is analysed by χ<sups>2</sups> tests.</p> <hd id="AN0146531279-7">Results</hd> <p>Table 1 shows the percentages of correct answers for each measurement (FM and ZM). This percentage was higher when the interviewer placed the object on the zero mark than when the child freely placed the object on the ruler to measure it (ZM, 79.8%, <emph>n</emph> = 79; FM, 33.3%, <emph>n</emph> = 33; McNemar, <emph>p</emph> < 0.00). Furthermore, although giving a correct measurement in the first trial is independent of measuring correctly in the second trial (χ<sups>2</sups> = 2.01, <emph>p</emph> = 0.157), there were 29 children who measured correctly in both situations. In contrast, 50 children measured correctly in only the Measurement from Zero, while only 4 measured correctly in only the Free Measurement. After a deeper analysis, 3 of them have been considered random answers, and the other one will be discussed further in the <emph>Discussion</emph> section.</p> <p>Percentage and frequency, % (F), of correct and wrong answers in Free Measurement and Measurement from Zero</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left"><p>ZM wrong</p></th><th align="left"><p>ZM correct</p></th></tr></thead><tbody><tr><td align="left"><p>FM wrong</p></td><td align="left"><p>24.2% (16)</p></td><td align="left"><p>75.8% (50)</p></td></tr><tr><td align="left"><p>FM correct</p></td><td align="left"><p>12.1% (4)</p></td><td align="left"><p>87.9% (29)</p></td></tr></tbody></table> </ephtml> </p> <p>N = 99</p> <p>Table 2 contains the percentage of correct answers for both measurements for each grade or age group. The older group got more correct answers than the rest of the groups for both measurements, and the 1st grade group had the lowest percentage of correct answers. However, the difference with the grade group is not significant in the Free Measurement (χ<sups>2</sups> (<reflink idref="bib2" id="ref60">2</reflink>, N = 99) = 4.84, <emph>p</emph> = 0.09), nor in the Measurement from Zero (χ<sups>2</sups> (<reflink idref="bib2" id="ref61">2</reflink>, N = 99) = 0.75, <emph>p</emph> = 0.75).</p> <p>Percentage and frequency, % (F), of correct answers for each grade</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Grade group</p></th><th align="left"><p>FM</p></th><th align="left"><p>ZM</p></th></tr></thead><tbody><tr><td align="left"><p>Kindergarteners</p></td><td align="left"><p>29.7% (11)</p></td><td align="left"><p>81.1% (30)</p></td></tr><tr><td align="left"><p>1st graders</p></td><td align="left"><p>21.4% (6)</p></td><td align="left"><p>75.0% (21)</p></td></tr><tr><td align="left"><p>2nd graders</p></td><td align="left"><p>47.1% (16)</p></td><td align="left"><p>82.4% (28)</p></td></tr><tr><td align="left"><p>Total</p></td><td align="left"><p>33.3% (33)</p></td><td align="left"><p>79.8% (79)</p></td></tr></tbody></table> </ephtml> </p> <p>N = 99</p> <p>When analysing the strategies used by the children, through the verbalisation of their answers, they tended to use the same strategy in the Measurement from Zero and the Free Measurement (χ<sups>2</sups> (<reflink idref="bib1" id="ref62">1</reflink>, N = 99) = 35. 07, <emph>p</emph> < 0.00). Upon closer analysis, the strategy used most frequently was reading of the endpoint (with 65 children using EP in FM and ZM), distantly followed by counting marks (with only 9 using CM in both measurements).</p> <p>However, when analysing correlations between stating a correct answer in the Measurement from Zero and the strategy used, the results show that they are associated (χ<sups>2</sups> (<reflink idref="bib2" id="ref63">2</reflink>, N = 99) = 43.95, <emph>p</emph> < 0.00) in such a way that when they read the endpoint they tended to give a correct answer. However, in the Free Measurement this relation was not found (χ<sups>2</sups> (<reflink idref="bib2" id="ref64">2</reflink>, N = 99) = 0.82, <emph>p</emph> = 0.67).</p> <p>Table 3 shows the strategies according to grade group. When children measured freely (FM), there was a relationship between the participants' grade group and the strategy employed (χ<sups>2</sups> (<reflink idref="bib2" id="ref65">2</reflink>, N = 99) = 9.51, <emph>p</emph> = 0.01) in such a way that kindergarteners and 2nd graders mainly read the endpoint while 1st graders read the endpoint and counted marks with similar frequencies. However, for the Measurement from Zero, the above association is not present (χ<sups>2</sups> (<reflink idref="bib2" id="ref66">2</reflink>, N = 99) = 0.29, <emph>p</emph> = 0.87). On the other hand, in all of the grade groups, reading of the endpoint was the most used strategy.</p> <p>Percentage<sups>a</sups> and frequency, % (F), for each strategy in the Free Measurement and the Measurement from Zero by grade</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="2"><p>Grade group</p></th><th align="left" colspan="3"><p>FM</p></th><th align="left" colspan="3"><p>ZM</p></th></tr><tr><th align="left"><p>EP</p></th><th align="left"><p>CM</p></th><th align="left"><p>OTH</p></th><th align="left"><p>EP</p></th><th align="left"><p>CM</p></th><th align="left"><p>OTH</p></th></tr></thead><tbody><tr><td align="left"><p>Kindergarteners</p></td><td align="left"><p>83.8% (31)</p></td><td align="left"><p>5.4% (2)</p></td><td align="left"><p>10.8% (4)</p></td><td align="left"><p>78.4% (29)</p></td><td align="left"><p>2.7% (1)</p></td><td align="left"><p>18.9% (7)</p></td></tr><tr><td align="left"><p>1st graders</p></td><td align="left"><p>46.4% (13)</p></td><td align="left"><p>35.7% (10)</p></td><td align="left"><p>17.9% (5)</p></td><td align="left"><p>71.4% (20)</p></td><td align="left"><p>21.4% (6)</p></td><td align="left"><p>7.1% (2)</p></td></tr><tr><td align="left"><p>2nd graders</p></td><td align="left"><p>76.5% (26)</p></td><td align="left"><p>11.8% (4)</p></td><td align="left"><p>11.8% (4)</p></td><td align="left"><p>79.4% (27)</p></td><td align="left"><p>8.8% (3)</p></td><td align="left"><p>11.8% (4)</p></td></tr><tr><td align="left"><p>Total</p></td><td align="left"><p>70.7% (70)</p></td><td align="left"><p>16.2% (16)</p></td><td align="left"><p>13.1% (13)</p></td><td align="left"><p>76.8% (76)</p></td><td align="left"><p>10.1% (10)</p></td><td align="left"><p>13.1% (13)</p></td></tr></tbody></table> </ephtml> </p> <p>N = 99 <sups>a</sups>The percentages are calculated on the total by separate rows for FM and ZM</p> <p>Table 4 shows the strategies used by the children who measured correctly, according to their grade group. As in the general analysis of the strategies, the most commonly used in both Free Measurement and Measurement from Zero was reading of the endpoint, exhibiting no association with grade group in any of the measurements (χ<sups>2</sups> (<reflink idref="bib2" id="ref67">2</reflink>, N = 33) = 2.93, <emph>p</emph> = 0.23, for FM; χ<sups>2</sups> (<reflink idref="bib2" id="ref68">2</reflink>, N = 79) = 2.23, <emph>p</emph> = 0.33, for ZM).</p> <p>Percentage<sups>a</sups> and frequency, % (F), of strategies in the correct answers in both measurements for each grade group</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="2"><p>Grade group</p></th><th align="left" colspan="3"><p>FM</p></th><th align="left" colspan="3"><p>ZM</p></th></tr><tr><th align="left"><p>EP</p></th><th align="left"><p>CM</p></th><th align="left"><p>OTH</p></th><th align="left"><p>EP</p></th><th align="left"><p>CM</p></th><th align="left"><p>OTH</p></th></tr></thead><tbody><tr><td align="left"><p>Kindergarteners</p></td><td align="left"><p>90.9% (10)</p></td><td align="left"><p>9.1% (1)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>83.3% (25)</p></td><td align="left"><p>3.3% (1)</p></td><td align="left"><p>13.3% (3)</p></td></tr><tr><td align="left"><p>1st graders</p></td><td align="left"><p>50.0% (3)</p></td><td align="left"><p>50.0% (3)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>95.2% (20)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>4.8% (1)</p></td></tr><tr><td align="left"><p>2nd graders</p></td><td align="left"><p>75.0% (12)</p></td><td align="left"><p>6.3% (1)</p></td><td align="left"><p>18.8% (3)</p></td><td align="left"><p>92.9% (26)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>7.1% (2)</p></td></tr><tr><td align="left"><p>Total</p></td><td align="left"><p>75.8% (25)</p></td><td align="left"><p>15.2% (5)</p></td><td align="left"><p>9.1% (3)</p></td><td align="left"><p>89.9% (71)</p></td><td align="left"><p>1.3% (1)</p></td><td align="left"><p>8.9% (7)</p></td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Percentages are calculated on the total by separate rows for FM and ZM</p> <p>In the Free Measurement, children measured without any assessment, so information on where they aligned the object and the ruler was analysed. The results of these relative object-ruler positions are shown in Table 5. It is surprising that almost half the sample (47.5%), mainly kindergarten children and 1st graders, placed the object on the 2 mark or beyond the 2 mark, while most of the participants placed the object at 0. On the other hand, there is a significant relationship between the placement of the object on the ruler and the grade group (χ<sups>2</sups> (<reflink idref="bib2" id="ref69">2</reflink>, N = 99) = 8.47, <emph>p</emph> = 0.01) in such a way that kindergarteners and 1st graders tended to place the object at the 2 mark or beyond and 2nd graders at the 0 mark.</p> <p>Percentage<sups>a</sups> and frequency, % (F), of situation of the object on the ruler in the Free Measurement by grade group</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Grade group/situation</p></th><th align="left"><p>− 1</p></th><th align="left"><p>0</p></th><th align="left"><p>1</p></th><th align="left"><p>2</p></th></tr></thead><tbody><tr><td align="left"><p>Kindergarteners</p></td><td align="left"><p>13.5% (5)</p></td><td align="left"><p>21.6% (8)</p></td><td align="left"><p>10.8% (4)</p></td><td align="left"><p>54.1% (20)</p></td></tr><tr><td align="left"><p>1st graders</p></td><td align="left"><p>10.7% (3)</p></td><td align="left"><p>7.1% (2)</p></td><td align="left"><p>21.4% (6)</p></td><td align="left"><p>60.7% (17)</p></td></tr><tr><td align="left"><p>2nd graders</p></td><td align="left"><p>17.6% (6)</p></td><td align="left"><p>44.1% (15)</p></td><td align="left"><p>8.8% (3)</p></td><td align="left"><p>29.4% (10)</p></td></tr><tr><td align="left"><p>Total</p></td><td align="left"><p>14.1% (16)</p></td><td align="left"><p>25.3% (23)</p></td><td align="left"><p>13.1% (13)</p></td><td align="left"><p>47.5% (47)</p></td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Percentages are calculated on the total by row</p> <p>The association between position and correct measurement is statistically significant (χ<sups>2</sups> (<reflink idref="bib3" id="ref70">3</reflink>, N = 99) = 52.69, <emph>p</emph> < 0.00), so that, with the data from Table 5, we can say that the children who placed the object at 0 tended to measure correctly and those who placed it at 2 mostly answered incorrectly.</p> <p>Since what determines the correct measurement is not only the strategy but also the object-ruler relative position, the study of this combination is crucial, particularly in the children who stated correct answers. The frequencies for the different combinations are shown in Table 6.</p> <p>Percentage<sups>a</sups> and frequency, % (F), of situation of the object in the ruler and strategy for correct answers in the Free Measurement</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Situation/strategy</p></th><th align="left"><p>EP</p></th><th align="left"><p>CM</p></th><th align="left"><p>OTH</p></th></tr></thead><tbody><tr><td align="left"><p>− 1</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>0% (0)</p></td></tr><tr><td align="left"><p>0</p></td><td align="left"><p>95.8% (22)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>4.3% (1)</p></td></tr><tr><td align="left"><p>1</p></td><td align="left"><p>100% (2)</p></td><td align="left"><p>0% (0)</p></td><td align="left"><p>0%(0)</p></td></tr><tr><td align="left"><p>2</p></td><td align="left"><p>12.5% (1)</p></td><td align="left"><p>62.5% (5)</p></td><td align="left"><p>25.0% (2)</p></td></tr><tr><td align="left"><p>Total</p></td><td align="left"><p>75.8% (25)</p></td><td align="left"><p>15.2% (5)</p></td><td align="left"><p>9.1% (3)</p></td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Percentages are calculated on the total by row</p> <p>Focusing on the children who measured correctly in the Free Measurement, we found an association between the strategy they used and the position at which they placed the object with respect to the ruler (χ<sups>2</sups> (<reflink idref="bib2" id="ref71">2</reflink>, N = 33) = 20.64, <emph>p</emph> < 0.00). Thus, 95.8% (F = 22) of those who placed the object on the 0-hash mark read the endpoint, while 62.5% (F = 5) of the children who placed the object on the 2 mark or beyond counted marks.</p> <hd id="AN0146531279-8">Discussion</hd> <p>The aim of this paper was to explore how children aged 5 to 8 use the conventional ruler, given that they had not received prior school instruction, with emphasis on the analysis of the object-ruler relative position and the strategy followed to verbalise the results, as well as the relationship between these by those who give correct answers. The possible influence that the grade level or age group has on the students' ability to obtain the correct measurement, together with the possible relation with the transition from kindergarten to primary school, was also studied. For this purpose, children from kindergarten and primary school were asked to measure a strip of cardboard with a conventional ruler freely and after aligning the left end of the cardboard strip at the zero mark of the ruler. In both cases, they were asked how long it was and how they obtained the answer (or, how they knew it). The results showed that, on the one hand, children at these ages found it difficult to measure correctly with a conventional ruler, especially when they had to do it without directions. On the other hand, in the free measurement, where the relative position of the object and ruler is crucial to giving a correct answer; positioning the cardboard strip at the ruler hash mark 2 or beyond was identified as the most frequently used; the endpoint reading as the strategy most used by children to justify their measurements and the combination of that strategy with the object placement at 0 was the most used by children who measured correctly.</p> <p>Besides working with no altered manipulatives and specifically asking children to verbalise their measurement strategies, another noteworthy aspect of this research project was its offering tasks to children involving the free measurement of an object focusing on the absence of school instruction on the use of the ruler. This task can provide a lot of information about the child's intuitive level of interpretation of the origin of the measurement and units in the instrument, although we could not control for whether or not they had received any instruction at home.</p> <p>The percentage of students who measured correctly in the Free Measurement, as shown here, is much lower than the one found by Nunes et al. ([<reflink idref="bib27" id="ref72">27</reflink>]) with children between the ages of 6–8. This difference could be due to the fact that our study included younger children, 5–6 years of age (3rd Kindergarten), whose correct measurement rate is the lowest, and this particular age group was not included in the aforementioned study. Our age range most closely resembles that of the McDonough and Sullivan study ([<reflink idref="bib24" id="ref73">24</reflink>]), which asked children between the ages of 4 and 7 to measure a 20 cm straw with a ruler, without specifying if school instruction had occurred. In this case, the correct measurement rate is similar to that of our study. Our results in the Free Measurement are also very similar to those of the study by Kotsopoulos et al. ([<reflink idref="bib20" id="ref74">20</reflink>]), who ask 4- and 5-year-olds to measure the two lines of a T and say which is longer.</p> <p>There is controversy about the association between age or grade level and measurements with a conventional ruler. Kotsopoulos et al. ([<reflink idref="bib20" id="ref75">20</reflink>]) found this association, but Nunes et al. ([<reflink idref="bib27" id="ref76">27</reflink>]) did not. In the present study, as well as in the one of Gómezescobar et al. ([<reflink idref="bib13" id="ref77">13</reflink>]), there is not significant relationship between age or grade group and success in free measurement. Therefore, in order to clarify, we recommend further research on this issue.</p> <p>As Nunes et al. ([<reflink idref="bib27" id="ref78">27</reflink>]) noted, we found that 2nd graders performed better than the 1st graders in primary school. It is also interesting to note that as with the work of Gómezescobar et al. ([<reflink idref="bib13" id="ref79">13</reflink>]), the correct measurement percentage was lower in the primary school 1st graders than in the oldest children in kindergarten.</p> <p>Despite the fact that in none of the grade levels had children received school instruction on how to measure with the ruler, we might consider that this decrease could be associated with the general difference in the instruction between these two stages, as reported by several authors (González et al. [<reflink idref="bib14" id="ref80">14</reflink>]; Tamayo [<reflink idref="bib32" id="ref81">32</reflink>]), stating that it is more globalised and constructivist in kindergarten than in primary school. The differences between these stages are also evidenced by the use of materials and the proposed tasks. In kindergarten, the materials and tasks are usually more manipulative and focused on everyday life, whereas in primary school the tasks are rather compartmentalised in specific fields, and the materials used are almost exclusively printed. Therefore, this stage transition is characterised by giving up manipulation, which could negatively affect how the children face a real measurement situation.</p> <p>Taking into account the nature of the tasks, as well as the order in which they were presented to the participants, they both involve manipulation, and the Free Measurement was performed before the Measurement from Zero. This fact could make us consider that the best results in Measurement from Zero could be due to a facilitation of the Free Measurement. However, there is no correlation between measuring correctly in the Free Measurement and measuring correctly in the Measurement from Zero, which makes these two events independent for the participants. One possible explanation for this fact can be found in analysing the strategies the students employed. Although a more extensive reflection on these strategies is carried out further on, at this point, we can highlight the tendency of children to use the same strategy in both measurements. This would show us that they establish an instrument (ruler)-strategy (reading of the endpoint) link regardless of the fact that they face different measurement situations with the ruler. Therefore, we can infer that these children demonstrate a poor conceptualisation of the origin of the length measurement when measuring with the ruler, so they exhibit the zero point difficulty, as was reported by Kamii ([<reflink idref="bib17" id="ref82">17</reflink>]) and Bragg and Outhred ([<reflink idref="bib4" id="ref83">4</reflink>]) in their studies. Besides that, Clements et al. ([<reflink idref="bib8" id="ref84">8</reflink>]) found that some children at the end of 3rd grade (8–9 years old) showed evidence of being at the <emph>length measuring level</emph> in their learning trajectory. Given that the children participating in this research study attended 3rd of kindergarten, 1st and 2nd grade of primary school—all the grades prior to the one mentioned by Clements and his collaborators—we could conclude the students of the present study are mostly at levels prior to the <emph>length measuring level</emph> in their conceptualisation of the ruler. This would justify the need for specific instruction on the use of such an instrument.</p> <p>With respect to the verbalised strategies, in our case, the most used strategy in both the Free Measurement and the Measurement from Zero is the reading of the endpoint. As noted above, children linked instrument and strategy, indiscriminately employing a strategy that works well only in one of the measurement situations, when the ruler and the object are aligned at zero (Levine et al. [<reflink idref="bib22" id="ref85">22</reflink>]). Therefore, the strategy used the most would work in the Measurement from Zero but not in the Free Measurement since it depends on where the children place the object on the ruler. The other two strategies used (counting marks and other) showed a similar percentage among them, and a considerably lower percentage than reading of the endpoint. In the cases where children stated a correct measure, the predominant strategy remained reading of the endpoint. This finding appears to contradict the findings reported by other authors, such as Cullen and Barrett ([<reflink idref="bib10" id="ref86">10</reflink>]), who worked with kindergarteners (4–5 years old) and 2nd graders (7–8 years old) and found reading of the endpoint to be the third strategy used for reporting the correct measure, preceded by counting intervals and counting marks. Nonetheless, in both studies, the measurement situations were completely different regarding the ruler and object alignment. Moreover, it is important noticing that Cullen and Barret ([<reflink idref="bib10" id="ref87">10</reflink>]) did not used conventional rulers but rather simplified 9-inch rulers with only inch marks (not subdivisions). Those conditions—the misalignment and the simplified ruler—could have led the students to mainly use a strategy related to the action on the interval, as they found. The action on the interval entails a great conceptual burden in terms of length measurement, because regardless of the situation of the object or the conditions of the measuring instrument, it focuses on the space interval and therefore evidences a child's clear perception of the unit on the ruler. Since in our work the most used strategy in correct measurements is the reading of the endpoint, we have no evidence of the participating children identifying the unit, although as they stated correctly, it may have been their insight, without being part of what they verbalised.</p> <p>On the other hand, Kamii ([<reflink idref="bib18" id="ref88">18</reflink>]), who worked with 3rd grade students (8–9 years old), found a fairly equal percentage between counting marks and reading of the endpoint strategies, 31% and 30%, respectively. It is a similar case to the findings of our study attributed to 1st graders in the Free Measurement. Provided that other studies (Congdon et al. [<reflink idref="bib9" id="ref89">9</reflink>]; Levine et al. [<reflink idref="bib22" id="ref90">22</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref91">30</reflink>]) suggested that children who counted marks were more prepared to identify the unit, which is one of the key aspects for understanding length measurements, the 1st graders of this study could be closer in understanding the role of the unit than those in the other grades. Consequently, we could say that children in the first year of primary school of this sample are more ready to make a more meaningful use of the ruler than children in the other two grades, contrary to what their results on obtaining the correct measures show. These 1st graders would be in what Barrett et al. ([<reflink idref="bib1" id="ref92">1</reflink>]) considers a necessary transition phase towards counting intervals, as they focus their attention on the marks and endpoints.</p> <p>The relative object-ruler position in the free measurement has also been analysed in this study. It gives us information about the origin or zero point interpretation that children make. As we emphasised in the introduction, we only found two previous studies where this variable was examined in real measurement situations with the conventional ruler, but these studies worked with students who were older than ours (Kamii [<reflink idref="bib17" id="ref93">17</reflink>]; Bragg and Outhred [<reflink idref="bib4" id="ref94">4</reflink>]). In this work, it was observed that the point of the ruler where children placed the object was related to the age or grade level in such a way that the children in the youngest group (kindergarten) tended to align the object from the 2 mark, while those in the oldest group (2nd primary school) tended to align the object at 0. In addition, the location of the object on the ruler and the correctness of the answer also associate, being that the alignment at 0 was the one that produced the highest percentage of correct measures. Aligned with that, for children who made correct measurements in the Free Measurement, the location of the object on the ruler and the strategy used to verbalise the result were also closely associated. In addition to that, children tended to count the hash marks when they placed the object at 2 or beyond the 2 mark, while those who placed the object at 0 read the endpoint.</p> <p>It is worth mentioning here the case of the two children who placed the cardboard strip close to the 1 mark. In these cases, the students read of the endpoint and gave the correct measurement, disregarding the lack of perfect alignment with the number to the right of the object's endpoint. When placing the object almost at 1 but shifting left, the number closest to the object's endpoint would be 8, even if the object overlaps it. This fact could be interpreted as a lack of development of measurement rigor in the child, which is in line with the findings of Castle and Needman ([<reflink idref="bib6" id="ref95">6</reflink>]), who found in their study that 6-year-olds did not evidence the need to measure objects accurately.</p> <p>The case we referred to earlier in the analysis of Table 1 is also highlighted here. This child placed the left end of the object at the 8 hash mark and verbalised "eight centimetres", so the object was considered to be measured correctly. However, in the Measurement form Zero, the child said that the object measured "zero centimetres". Therefore, the meaningful lining up for this child was the left end alignment, which makes us conclude that it is the only case where the strategy that Cullen and Barrett ([<reflink idref="bib10" id="ref96">10</reflink>]) identified as left endpoint was used.</p> <p>The difficulties of children in the first years of school when using a conventional measurement tool could be related to the emphasis given in the classroom to numerical skills, as McDonough and Sullivan ([<reflink idref="bib24" id="ref97">24</reflink>]) pointed out, rather than related to problems of reading numbers, manipulation of numbers in general or the ability to operate with images. As a consequence of this emphasis on numerical skills, children could consider the role of numbers on the ruler to be the same as when they are doing other arithmetical tasks. For example, placing the object at the 1 mark could be related to the fact than 1 is the first number they use when they count. Then, we agree with Irwin et al. ([<reflink idref="bib16" id="ref98">16</reflink>]) that children do not distinguish the difference between the use of the number in the numerical series nor its use in measuring instruments. A possible cause may be due to the school emphasis on mathematical tasks related to arithmetic (McDonough and Sullivan [<reflink idref="bib24" id="ref99">24</reflink>]).</p> <p>The children's answers when verbally reporting their measurement show that they thought that they should give a number as the answer to the question, "How long is it?" We observed that in order to give an answer, the children made use of their knowledge about numbers, especially as a numerical series. With respect to the interpretation of numbers on the ruler, the findings of other studies (Gómezescobar et al. [<reflink idref="bib13" id="ref100">13</reflink>]; Solomon et al. [<reflink idref="bib30" id="ref101">30</reflink>]) showed that numbers hinder the conceptualisation of the interval as an iterable object, a necessary concept for understanding the ruler structure. Concerning the unit identification on the ruler, in this study, none of the participants used the counting interval strategy (or units), so we deduced that the students did not identify the unit.</p> <p>To conclude, the participants in this study, children in the third year of kindergarten and the first and second years of primary school, aged 5 to 8 years, generally do not measure correctly when they are given an 8 cm strip to be measured with a conventional ruler. The results improved significantly when they received additional help aligning the strip to the zero point of the same ruler. Regarding the strategies used to verbally report the measurement, children mainly used the endpoint reading strategy, even though they did not tend to place it at the 0 hash mark. Finally, the only children who measured correctly were those who lined the cardboard strip up at the 0 point of the ruler and used the EP strategy.</p> <p>In this study, it has been shown that, despite the fact that the ruler is a familiar instrument for children, its correct use is neither natural nor intuitive. The structure of the ruler, as a length measurement instrument, implies the interpretation of the zero point and the identification of the unit, its iteration and accumulation, without that unit being evident to or easily detectable by the child. Therefore, the correct use of this instrument requires specific instruction (Kellman and Massey [<reflink idref="bib19" id="ref102">19</reflink>]; McDonough and Sullivan [<reflink idref="bib24" id="ref103">24</reflink>]). The ruler marks and their corresponding numbers indicate the distance accumulation resulting from the iteration of the unit, in this case the centimetre, which is not naturally perceived by children. It is therefore necessary that the meaning of the hash marks, intervals and numbers be integrated into instruction in order for the student to reach the level of <emph>Consistent Length Measurer</emph> (Barrett et al. [<reflink idref="bib1" id="ref104">1</reflink>]).</p> <p>Therefore, in line with Kamii ([<reflink idref="bib18" id="ref105">18</reflink>]), instead of reducing the teaching of measuring lengths with a ruler to a procedural instruction, it is proposed that logical reasoning and conceptual exploration be fostered in children under the age of 8, since it seems from our results that these children are at a level prior to the <emph>Conceptual Ruler Measurer</emph> (Clements et al. [<reflink idref="bib8" id="ref106">8</reflink>]). Hence, it would seem interesting to begin the study of measuring with a ruler by focusing on the identification of the unit, which are the intervals and the origin of measurements. It would also be appropriate to offer students a manipulative that helps display the units on the ruler as objects. To this end, it is also proposed that, prior to using a conventional ruler, specially prepared rulers including discrete units within the continuous measurement tool be used for obtaining measurements (Gómezescobar et al. [<reflink idref="bib13" id="ref107">13</reflink>]). The first stage could be to help children construct a ruler where specific objects to be measured just fit within the intervals, with marks delimiting them; then include the numbers, which indicate the interval amounts (not on the marks, which are typically on a conventional ruler) and the accumulation of the distance. When these numbered objects that are placed within the intervals are removed, the units would be represented by consecutive marks, which, when numbered, would have a structure similar to that of the conventional ruler.</p> <hd id="AN0146531279-9">Funding</hd> <p>Funding was provided by Fundación Española para la Ciencia y la Tecnología, FECYT (Grant No. FCT-16-10952).</p> <hd id="AN0146531279-10">Acknowledgements</hd> <p>The authors would like to thank Dr. Ignacio Rieiro for his collaboration in the review of this paper. The first author acknowledges the Spanish Ministry of Education, Culture and Sports for her Erasmus Practice scholarship.</p> <hd id="AN0146531279-11">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0146531279-12"> <title> References </title> <blist> <bibl id="bib1" idref="ref16" type="bt">1</bibl> <bibtext> Barrett JE, Sarama J, Clements DH, Cullen C, McCool J, Witkowski-Rumsey C, Klanderman D. Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study. Mathematical Thinking and Learning. 2012; 14; 1: 28-54. 10.1080/10986065.2012.625075</bibtext> </blist> <blist> <bibl id="bib2" idref="ref22" type="bt">2</bibl> <bibtext> Boulton-Lewis GM, Wilss LA, Mutch SL. An analysis of young children's strategies and use of devices for length measurement. 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  Data: How Long Is It? Difficulties with Conventional Ruler Use in Children Aged 5 to 8
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Data: The ruler is the standard measuring instrument used for measuring lengths. However, measuring lengths with a ruler is a challenge for children. For this reason, this study explores how children who have not received specific school instruction on its use, measure lengths with a conventional ruler. The relative object-ruler position, the strategy to justify the measurement, and the combination between them are analysed. Additionally, the possible influence of the grade and the transition from Kindergarten to Primary School is also studied. To achieve this aim, 99 children were asked to measure a cardboard strip in both a free and a directed situation. The results showed that in free measurements children tend to situate the object in the 2 hash mark of the ruler, the reading of the endpoint was identified as the most used strategy, and the combination of this strategy with lining up the object at 0 was the most commonly used in correct measurements. On the other hand, the results also showed marginal significant differences between age groups in such a way that children in the last year of kindergarten measured better than those in the first year of primary school. To conclude, the educational implications of these results are discussed.
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