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Rational Number and Proportional Reasoning in Early Secondary School: Towards Principled Improvement in Mathematics

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Title: Rational Number and Proportional Reasoning in Early Secondary School: Towards Principled Improvement in Mathematics
Language: English
Authors: Howe, Christine, Luthman, Stefanie, Ruthven, Kenneth, Mercer, Neil, Hofmann, Riikka, Ilie, Sonia, Guardia, Paula
Source: Research in Mathematics Education. 2015 17(1):38-56.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 19
Publication Date: 2015
Document Type: Journal Articles
Reports - Research
Education Level: Secondary Education
Descriptors: Foreign Countries, Mathematical Concepts, Numbers, Secondary School Mathematics, STEM Education, Mathematics Instruction, Science Instruction, Mathematics, Secondary School Science, Fractions, Statistical Analysis, Control Groups, Comparative Analysis, Pretests Posttests, Thinking Skills, Mathematical Aptitude, Problem Solving, Likert Scales, Questionnaires, Multiple Regression Analysis
Geographic Terms: United Kingdom
DOI: 10.1080/14794802.2015.1019914
ISSN: 1479-4802
Abstract: Reflecting concerns about student attainment and participation in mathematics and science, the Effecting Principled Improvement in STEM Education ("epiSTEMe") project attempted to support pedagogical advancement in these two disciplines. Using principles identified as effective in the research literature (and combining these in a novel fashion), the project developed topic modules for early secondary-school teaching in the UK, arranged for their implementation in classrooms, and evaluated the results. This article reports the development, implementation and evaluation of the "epiSTEMe" mathematics module entitled "Fractions, Ratios and Proportions." The module covers aspects of rational number and proportional reasoning relevant to the early secondary curriculum, and was developed in collaboration with teachers, implemented in 11 classrooms, and evaluated through comparison with 16 control classrooms where the topic was addressed using established methods. Students who used the "epiSTEMe" materials made significantly greater progress than control students as regards topic mastery, while holding positive opinions about their teaching and learning experiences.
Abstractor: As Provided
Number of References: 44
Entry Date: 2016
Accession Number: EJ1091428
Database: ERIC
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  Value: <anid>AN0101854577;[46o1]01mar.15;2019Feb11.14:18;v2.2.500</anid> <title id="AN0101854577-1">Rational number and proportional reasoning in early secondary school: towards principled improvement in mathematics. </title> <p>Reflecting concerns about student attainment and participation in mathematics and science, the Effecting Principled Improvement in STEM Education (epiSTEMe) project attempted to support pedagogical advancement in these two disciplines. Using principles identified as effective in the research literature (and combining these in a novel fashion), the project developed topic modules for early secondary-school teaching in the UK, arranged for their implementation in classrooms, and evaluated the results. This article reports the development, implementation and evaluation of the epiSTEMe mathematics module entitled Fractions, Ratios and Proportions. The module covers aspects of rational number and proportional reasoning relevant to the early secondary curriculum, and was developed in collaboration with teachers, implemented in 11 classrooms, and evaluated through comparison with 16 control classrooms where the topic was addressed using established methods. Students who used the epiSTEMe materials made significantly greater progress than control students as regards topic mastery, while holding positive opinions about their teaching and learning experiences.</p> <p>Keywords: rational number; proportional reasoning; classroom dialogue</p> <hd id="AN0101854577-2">Introduction</hd> <p>The effectiveness of school education in mathematics and science is an enduring concern (Gilbert, [<reflink idref="bib13" id="ref1">13</reflink>]; Roberts, [<reflink idref="bib34" id="ref2">34</reflink>]), with attainment and post-compulsory participation both spotlighted as significant issues. In the UK, concerns about attainment are often linked to the nation's middling and relatively static position in international league tables (e.g. PISA, [<reflink idref="bib32" id="ref3">32</reflink>], [<reflink idref="bib33" id="ref4">33</reflink>]; TIMSS, [<reflink idref="bib42" id="ref5">42</reflink>], [<reflink idref="bib43" id="ref6">43</reflink>]). As regards post-compulsory participation, the Royal Society ([<reflink idref="bib35" id="ref7">35</reflink>]) highlights that only 17.3% of "the potentially eligible population" in the UK is enrolled for mathematics and/or core science. Moreover amongst participants, there are marked asymmetries across student sub-groups. For instance, reflecting long-standing disparities, boys accounted for 61% of the entrants for mathematics in recent A levels and 79% of the entrants for physics (JCQ, [<reflink idref="bib22" id="ref8">22</reflink>]). This is despite the fact that girls have for some time performed at least as well as boys in these subjects at earlier stages. (A level is the main post-compulsory qualification across most of the UK).</p> <p>With this situation as backcloth, the UK's <emph>Economic and Social Research Council</emph> and <emph>Institute of Physics</emph> launched a programme of research concerned with attainment and participation in mathematics and science. The programme, which came to be known as the <emph>Targeted Initiative on Science and Mathematics Education,</emph> comprised five projects, some highlighting mathematics, some science and some both. Two of the projects with a mathematical focus independently identified rational number as a unique challenge, for this is a topic that does not simply cause difficulties when covered in school (Nunes & Bryant, [<reflink idref="bib30" id="ref9">30</reflink>]; Sophian, [<reflink idref="bib40" id="ref10">40</reflink>]). It is also associated with misconceptions that can endure into adulthood, even over basic issues like equivalence or difference of magnitude (DeWolf & Vosniadou, [<reflink idref="bib9" id="ref11">9</reflink>]; Schneider & Siegler, [<reflink idref="bib38" id="ref12">38</reflink>]). As Schneider and Siegler ([<reflink idref="bib38" id="ref13">38</reflink>]) indicate, "adults' performance with fractions resembles elementary school children's performance with whole number" (p. 1236). Echoing this, the National Mathematics Advisory Panel ([<reflink idref="bib29" id="ref14">29</reflink>]) in the US (p. 17) has declared that "the teaching of fractions must be acknowledged as critically important and improved", while policy documents in the UK have identified ratios and percentages as particular concerns (e.g. Scottish Executive, [<reflink idref="bib39" id="ref15">39</reflink>]). For all these reasons, charting attainment with rational number (plus the implications for participation) and/or developing pedagogic strategies to improve the situation became a major theme within the <emph>Targeted Initiative</emph>. The present article reports work with the latter focus (see Hodgen, Küchemann, Brown, & Coe [[<reflink idref="bib16" id="ref16">16</reflink>]] for an example that emphasises the former).</p> <hd id="AN0101854577-3">epiSTEMe and rational number</hd> <p>The reported research is part of the Effecting Principled Improvement in STEM Education (<emph>epiSTEMe</emph>) project, whose starting point was the major contribution that classroom processes are known to make to student outcomes (e.g. Hattie, [<reflink idref="bib15" id="ref17">15</reflink>]). The project attempted to: (<reflink idref="bib1" id="ref18">1</reflink>) use research literature to identify optimal processes; (<reflink idref="bib2" id="ref19">2</reflink>) distil conclusions in pedagogical principles for mathematics and science; (<reflink idref="bib3" id="ref20">3</reflink>) implement the principles during the early years of what, in the UK, is termed 'secondary education'; and (<reflink idref="bib4" id="ref21">4</reflink>) evaluate the impact. It realised these aims through teaching modules relating to rational number, a further topic in mathematics, and two topics in science. Focusing upon the early years of secondary school, evaluation revolved around attainment rather than post-compulsory participation. However, it included attitudinal measures as well as indices of attainment, and the implications for student subgroups were examined, including for those whose participation is a concern. Thus the reported research addresses the implications of the rational number module for student knowledge and associated attitudes, taking sub-group membership into account.</p> <p>As regards pedagogical principles, proportional reasoning emerged from the literature as a potentially productive context for addressing the challenges. While a grasp of rational number has been regarded as pre-requisite for proportional reasoning (e.g. Behr, Lesh, Post, & Silver, [<reflink idref="bib3" id="ref22">3</reflink>]; Charalambous & Pitta-Pantazi, [<reflink idref="bib7" id="ref23">7</reflink>]), the relation now appears to be reciprocal. Successful proportional reasoning supports mastery of rational number (Ben-Chaim, Fey, Fitzgerald, Benedetto, & Miller, [<reflink idref="bib4" id="ref24">4</reflink>]; Howe, Nunes, & Bryant, [<reflink idref="bib21" id="ref25">21</reflink>]; Lamon, [<reflink idref="bib23" id="ref26">23</reflink>]), indicating that teaching in the two areas should be aligned. Such alignment is not typically prioritised in UK classrooms, but it became a key principle underpinning the <emph>epiSTEMe</emph> module.</p> <p>It was recognised that in seeking alignment the module would inevitably place heavy emphasis on ratios, for proportional reasoning revolves around ratios. It entails "recognition of the constant ratio between elements of the same measure space and recognition of the functional relationship between measure spaces" (Lamon, [<reflink idref="bib23" id="ref27">23</reflink>], p. 638), e.g. recognition that as size differences between illuminated objects increase (ratio between elements) so size differences between their shadows also increase (relationship between measure spaces). Focusing upon ratios also amounts to a significant departure from standard UK practice, for the emphasis is typically upon fractions. Fractions are usually introduced early in primary school (often via partitioning of shapes), and become the lynchpins for subsequent coverage of decimals and percentages. Ratios are often left until last, and sometimes treated in isolation. Yet from an information processing perspective, ratios are the simplest form of rational number for uniquely they depend upon binary relations (English & Halford, [<reflink idref="bib12" id="ref28">12</reflink>]): saying that two spoons of oil are needed in vinaigrette for every spoon of vinegar requires comparison across two sets. Fractions by contrast depend upon tertiary relations, between the total of three spoons, the two spoons of oil, and the one spoon of vinegar. Moreover as Confrey, Maloney, Nguyen, Mojica, and Myers ([<reflink idref="bib8" id="ref29">8</reflink>]) indicate, ratios are implicit within fractions, "lurk[ing] in the background" (p. 350) whenever fractions are used. All in all, there are strong grounds for emphasising ratios, and this became a second principle behind the <emph>epiSTEMe</emph> module.</p> <p>At the same time, the module was also informed via general principles, which applied across the <emph>epiSTEMe</emph> project as a whole. The central principle amongst these (see Ruthven et al., [<reflink idref="bib36" id="ref30">36</reflink>], for others) emphasised small group and whole-class interaction in which students make extended contributions and consider contrasting perspectives in a reasoned fashion. Target knowledge emerges through productive resolution of differences, with teachers offering guidance rather than imposing authority. This form of interaction has come to be known as "dialogic teaching" (Alexander, [<reflink idref="bib1" id="ref31">1</reflink>]), and it has been used successfully in mathematics with students using talk effectively as the tool for reasoning and showing enhanced academic attainment (Mercer & Sams, [<reflink idref="bib26" id="ref32">26</reflink>]). Nevertheless, while previous research recommends dialogic teaching in general terms, it does not relate specifically to proportional reasoning and/or a ratio-centred approach, hence the need for the implementation and evaluation that, as noted, typified <emph>epiSTEMe</emph> as a whole. After describing the rational number module in detail, the sections that follow report and assess its implementation in UK schools.</p> <hd id="AN0101854577-4">Module development</hd> <p></p> <hd id="AN0101854577-5">Module structure</hd> <p>The module, entitled <emph>Fractions, Ratios and Proportions</emph> (abbreviated hereafter to <emph>Ratios</emph>), was developed over a two-year period in collaboration with eight teachers from secondary schools located near the university where the research team was based. Teachers contributed ideas about module content during full-day workshops held at the university (eight workshops in total), and between workshops piloted activities with their students and reported on effectiveness. In its final form, the <emph>Ratios</emph> module comprised five core lessons and two optional (and relatively challenging) extension lessons. Optional homework was provided. Lessons were notionally each of 50 minutes duration, but divided into two, three or four parts to facilitate adjustment to teaching periods of differing lengths. The module was detailed in teaching notes, with materials for classroom presentation also provided, i.e. PowerPoint slides, student booklets and worksheets, and a domino task. Materials are accessible via <ulink href="http://www.educ.cam.ac.uk/research/projects/episteme/epiSTEMeFractionsTNweb.pdf">http://www.educ.cam.ac.uk/research/projects/episteme/epiSTEMeFractionsTNweb.pdf</ulink></p> <p>Lesson 1 began by using real-world examples to introduce the concept of proportionality, i.e. through requiring identification of proportional terms in text and convergence on a working definition. It then highlighted the multiplicative nature of proportional reasoning via activities like <emph>Orange squash</emph> (see Figure 1), where correct statements had to be identified from options. Further practice with proportional reasoning was provided at the start of Lesson 2, with some problems covering rates (as defined in Ben-Chaim, Keret, & Ilany, [<reflink idref="bib5" id="ref33">5</reflink>]), e.g. distance travelled as a function of time to apply brakes. Thereafter the focus shifted to mathematical representation. Moving from left to right across slides like <emph>Squash lab</emph> (Figure 1), ratios were highlighted as inherent in all preceding activities, with fractions, percentages and decimals discussed as alternatives. Emphasis was placed on the relationship between ratios and the other forms, e.g. the part-part nature of the former compared with the part-whole nature of the latter. Moseley ([<reflink idref="bib28" id="ref34">28</reflink>]; see also Confrey et al., [<reflink idref="bib8" id="ref35">8</reflink>]; Howe et al., [<reflink idref="bib21" id="ref36">21</reflink>]; Nunes & Bryant, [<reflink idref="bib30" id="ref37">30</reflink>]) demonstrates the power of a comparative approach over part-whole analysis alone, even for fractions, percentages and decimals. Much of Lesson 3 was devoted to practice with the different forms, e.g. via a <emph>Pancake</emph> exercise (Figure 1), with the everyday application of ratios, fractions, percentages and decimals considered in conclusion, i.e. where each form is typically used and why it is preferred over alternatives. One slide (Figure 1) highlighted the historical dimension.</p> <p>Graph: Figure 1. Slides from Lessons 1–3.</p> <p>As indicated in Figure 1's <emph>Pancake</emph> exercise, Lesson 3 stressed independent computation of ratios, fractions, percentages and decimals. Using slides like <emph>Stickers</emph> (see Figure 2), the focus of Lesson 4 shifted towards direct conversion of one representation into another and computation of whole numbers from proportional information. Nevertheless, both lessons continued to highlight relations across different forms of representation. It was not until Lesson 5 that relations within the same form were addressed explicitly, i.e. the equivalence of 12:8 and 30:20, 9/12 and 3/4. Late treatment of this type of equivalence was a natural consequence of the ratio-centred approach, but it also responds to the high information load that the type involves (i.e. quaternary relations, English & Halford, [<reflink idref="bib12" id="ref38">12</reflink>]). The teaching notes indicated that equivalence within forms was likely to crop up earlier, e.g. recognition during the <emph>Pancake</emph> exercise that 3/5 of the mixture was milk as well as 6/10. However, teachers were encouraged to respond lightly and specifically, using such instances to set the scene for generalised treatment during Lesson 5.</p> <p>Graph: Figure 2. Slides from Lessons 4–7.</p> <p>Generalised treatment was supported via slides like <emph>Vinaigrette equivalence</emph> (Figure 2), and dominos where each tile displayed different values in their right and left segments, e.g. 1/27 and 6/50, 3/25 and 30/100. The aim was to form sequences such that the right segment of the first tile was equivalent to the left segment of the next tile, e.g. 6/50 and 3/25. Optional Lessons 6 and 7 consolidated earlier themes via harder problems, sometimes but not always requiring calculators. For instance <emph>Trips to school</emph> (Figure 2) involved computing percentages from decimals that were obtained using calculators, while <emph>Sorting numbers</emph> (also Figure 2) could be addressed without calculators. The final slides revisited proportional reasoning with more challenging values than in Lesson 1, e.g. having described Harry Potter as 1.68 metres tall and owning a wand of 38.1 cm, asking what size the giant Hagrid's wand would have to be to keep in proportion to Harry's when Hagrid is 1.22 metres taller than Harry. Again, rates were used along with other proportional relations.</p> <p>In sum, the lessons were fully compliant with the <emph>epiSTEMe</emph> principle of integrated treatment of proportional reasoning and rational number. As such, they spotlighted ratios, treating fractions, decimals and percentages contrastively with ratios (and with each other). Furthermore, all lessons respected the <emph>epiSTEMe</emph> principle of supporting high quality dialogue in small group and whole-class settings. For instance, it was suggested via the teaching notes that after deciding independently whether they agreed or disagreed with each statement in the <emph>Orange Squash</emph> problem (see Figure 1), students should discuss the statements in small groups and collectively identify the correct one. Group decisions and the reasoning behind decisions should be collated in whole-class plenary sessions, and differences (over decisions and reasoning) discussed and resolved. It was also suggested that application should be addressed at the end of Lesson 3 through whole-class discussion, with students brainstorming over the contexts where ratios, fractions, percentages and decimals are typically used, why in some contexts one form is preferred over alternatives, and how cultures managed historically when some forms were unavailable.</p> <hd id="AN0101854577-6">Evaluation instruments</hd> <p>Concurrently with module design, an opinion questionnaire and tests of knowledge were developed for purposes of evaluation. The opinion questionnaire was generic across the <emph>epiSTEMe</emph> modules, and is detailed in Ruthven et al. ([<reflink idref="bib37" id="ref39">37</reflink>]). It was printed on single A4 sheets for written completion immediately after the module's final lesson and comprised 10 pairs of 7-point Likert scales as shown in Table 1. The knowledge tests were specific to the <emph>Ratios</emph> module and comprised seven sets of problems, with Sets 1, 3, 5 and 7 emphasising proportional reasoning. Set 1 (two problems) focused upon mixing drinks at a children's party. For instance, Julia was said to use 2 parts orange juice and 5 parts water, and Jasmine to use 3 parts orange juice and 8 parts water. The task was to indicate whether the drinks tasted the same, Julia's tasted stronger, or Jasmine's tasted stronger. Set 3 (four problems) stated the ingredients needed to cook for eight people and requested the ingredients needed for six people, e.g. when pancakes for eight people require 8 tablespoons of flour, 16 tablespoons of milk, 2 tablespoons of beaten egg, and 1 teaspoon of salt. Set 5 (two problems) explained that thinking distance when stopping a car is directly related to the car's speed, and displayed thinking distance at 20mph and 40mph. The task was to compute values for further speeds, e.g. 60mph and 90mph. Set 7 (four problems) related to eels who were 5cm, 10cm and 15cm in length and whose diet had to be proportionate to length. One problem asked how many sprats should be fed to the shortest and longest eels when the middle-sized eel received 12 sprats.</p> <p>Table 1. Format of opinion questionnaire.</p> <p> <ephtml> <table><thead><tr><td>Scale</td><td>Illustrative item</td></tr></thead><tbody><tr><td>Pitch</td><td>These lessons were too difficult for me</td></tr><tr><td>Interest</td><td>Learning about this topic has been interesting</td></tr><tr><td>Effort</td><td>I worked hard in the lessons on this topic</td></tr><tr><td>Thinking</td><td>These lessons have made me think a lot</td></tr><tr><td>Future</td><td>I hope we don't study this topic again</td></tr><tr><td>Learning</td><td>These lessons have taught me a lot about this topic</td></tr><tr><td>Understanding</td><td>These lessons have helped me make sense of this topic</td></tr><tr><td>Explanation</td><td>I've got better at explaining things through taking part in these lessons</td></tr><tr><td>Value</td><td>These lessons have shown me why it's important to study this topic</td></tr><tr><td>Application</td><td>These lessons have helped me see how this topic applies to real life</td></tr></tbody></table> </ephtml> </p> <p>Rational number was highlighted more explicitly in Sets 2, 4 and 6. Set 2 (five problems) related to surveys, e.g. "Students were asked if they prefer watching television, listening to the radio or playing CDs. Of 72 students, 36 prefer television, 27 prefer radio, and 9 prefer CDs". The task was to judge the truth or falsity of statements like "Students prefer radio to CDs by a ratio of 3:1" and "9% of the students prefer listening to CDs". Set 4 (two problems) used items like "A bag contains red and green counters. 1/3 of the counters are red and 10 counters are green. How many red counters are in the bag? What is the ratio of red to green counters?" Three problems in Set 6 presented pairs of numbers, e.g. 3/4 and 0.80 vs 7/8 and 0.89. The concept of a number line was highlighted, and the task was to identify the pair that would be closest if placed on a line. A further three problems requested insertion of numbers into fractions to make statements correct, e.g. "0.25 equals _/32", "/20 is greater than 0.75". The tests were presented in illustrated booklets in which responses were to be written.</p> <p>There were three knowledge tests: (<reflink idref="bib1" id="ref40">1</reflink>) pre-test, for administration during the first lesson to assess initial understanding; (<reflink idref="bib2" id="ref41">2</reflink>) immediate post-test, for administration directly after the final lesson to assess understanding upon module completion; and (<reflink idref="bib3" id="ref42">3</reflink>) deferred post-test, for administration about one month later, to assess gain over a substantial period that also predicts longer-term performance (e.g. after 18 months [[<reflink idref="bib44" id="ref43">44</reflink>]]). Tests were compiled from a large battery that contained several versions of each set, e.g. an onion soup recipe in Set 3 as well as pancakes. The battery was administered to over 100 students from first-year secondary classes (not also involved in module development) whose proportionate accuracy per problem (number of students succeeding divided by number of students responding) provided indices of problem (hence set) difficulty. Selecting sets with reference to these indices, it was possible to ensure that the three tests were of equivalent difficulty while also allowing variation across tests (desirable to alleviate boredom). In fact, there was between 40% and 56% overlap across each pair of tests, with no version of each set appearing in more than two tests.</p> <p>Towards the end of the development period, four teachers who had contributed to the design implemented the module with first-year students and administered the tests to the planned schedule (i.e. start-of-module, end-of-module, after one month), as did two further teachers with equivalently aged classes in Scotland. The average effect size for student gain was +0.74 (Cohen's d) for pre- to immediate change and +0.98 for pre- to deferred change. This compared with values of -0.03 and -0.61 respectively from students in an informal comparison group, i.e. students from the same schools whose teachers were not involved with <emph>epiSTEMe</emph> and therefore covered the topic using established methods. Given the exercise's informality, the apparent decrement in the comparison group must be treated with caution. Nevertheless, it serves as a reminder that, as noted, the area is associated with enduring misconceptions, and lack of success with established methods is actually to be expected. In this context, the main function of the comparison group was to differentiate genuine progress after the <emph>Ratios</emph> module from practice effects due to repeated testing, not to inform about traditional approaches per se. This was also an important (and simplifying) assumption behind the large-scale implementation and formal evaluation. Emphasis was placed upon involving control students who would not follow the module but would complete the evaluation instruments at roughly the same time intervals as <emph>epiSTEMe</emph> students. While control teaching should address the material in broad terms to make instrument completion meaningful (and an attempt was made to ensure this), its precise match with the <emph>Ratios</emph> module was of lesser significance.</p> <hd id="AN0101854577-7">Large-scale implementation</hd> <p></p> <hd id="AN0101854577-8">Module implementation</hd> <p>Large-scale implementation began with an approach to all secondary schools within a <emph>c.,</emph>50 mile radius of the university (and therefore in England rather than across the UK). Information was provided about the project, together with an invitation to nominate up to two mathematics teachers who would be involved with first year classes (Year 7 in England). No specific guidance was offered over selection apart from recommending that the majority of students in relevant classes should have achieved Level 4 or above for mathematics in end-of-primary-school assessments.[<reflink idref="bib1" id="ref44">1</reflink>] Nominated teachers were invited to a briefing session, where the implications of participation were explained. It was emphasised that only 50% of participants would be given the <emph>Ratios</emph> module during the following year, i.e. constitute the intervention group. The remainder would be asked to act as controls, which would involve teaching fractions, ratios and proportions as usual, while administering the <emph>epiSTEMe</emph> evaluation instruments. It was stressed that after a 12-month interval, control teachers would receive module materials and be offered support over implementation that was equivalent to the intervention group. Schools whose teachers agreed to proceed were listed in order of their most recent CVA2-4 score.[<reflink idref="bib2" id="ref45">2</reflink>] In the hope of cross-condition equivalence, adjacently listed pairs of schools were assigned randomly to the intervention and control conditions. Thus, when two teachers volunteered from a single school, both were placed in the same condition.</p> <p>Implementation in the intervention schools was supported through two full-day professional development sessions for teachers, intentionally mimicking the support that is typically provided in England for educational innovation and so permitting assessment of effectiveness under normal circumstances. The first session (taking place before, or early in, the school year) focused upon dialogic teaching. Target practices were discussed and illustrated through video-extracts, and teachers were given activities to try with their students. Based on prior research (see Mercer & Littleton, [<reflink idref="bib25" id="ref46">25</reflink>]), some activities supported teachers in working with students to formulate 'ground rules' for generating productive small-group interaction, while others raised teachers' awareness of how they themselves talk in whole-class sessions. Further activities involved students in practising skills with mathematically relevant tasks and scoring talk for, e.g., <emph>listening carefully</emph> and <emph>explaining ideas</emph>. Teachers were encouraged to implement the activities with the Year 7 class they would be involving in <emph>epiSTEMe</emph> prior to the next professional development session, i.e. to establish dialogic methods before employing the modules.</p> <p>The second professional development session (held several weeks into the school year) began with discussion of how the dialogic work was progressing and, if necessary, trouble-shooting. It then moved to detailed introduction of the <emph>Ratios</emph> module (and the other <emph>epiSTEMe</emph> mathematics module). Teachers were given copies of the opinion questionnaire and knowledge tests, and asked to present these to their students at the designated points. Thereafter, contact was via email and telephone and, apart from periodic reminders, reactive to queries rather than proactive. The only exception was researcher visits to observe a sample of <emph>epiSTEMe</emph> lessons. Given the novelty of using dialogic methods in this area, it seemed important to document how far target practices were adopted. Observations were made using a nine-category schedule, covering behaviours (detailed later) that are central to dialogic teaching. The researcher observed for successive six-minute periods across full lessons, ticking on checklists to indicate which categories occurred within each period. She focused upon whole-class interaction given the logistical difficulties of observing group work. Before observing in classrooms, the researcher and a colleague independently coded videotapes recorded during the project's first two years, and established that the schedule could be used with adequate reliability (mean inter-judge agreement across categories = 72%).</p> <p>Control teachers were sent copies of the opinion questionnaire and knowledge tests by post at roughly the time that these instruments were given to the intervention group. They were asked to administer the instruments to the planned schedule around the block of lessons that best matched – in terms employed by the statutory curriculum then in force – "rational numbers, their properties and their different representations; applications of ratio and proportion" Control and intervention teachers were also asked to provide contextual information using instruments detailed in Ruthven et al. ([<reflink idref="bib37" id="ref47">37</reflink>]). These included an attitude questionnaire to be presented to students prior to relevant teaching (handed to intervention teachers at the first professional development session; posted concurrently to control teachers). The questionnaire's function was to establish pre-existing attitudes that might bear upon opinions or knowledge gains, and it comprised 20 7-point Likert scales, i.e. five scales relating to each of: (<reflink idref="bib1" id="ref48">1</reflink>) Ability in mathematics, e.g. <emph>I'm good at maths</emph>; (<reflink idref="bib2" id="ref49">2</reflink>) Enthusiasm for mathematics, e.g. <emph>Maths is boring</emph>; (<reflink idref="bib3" id="ref50">3</reflink>) Prospective involvement in mathematics, e.g. <emph>I'd like a job that involves using maths</emph>; and (<reflink idref="bib4" id="ref51">4</reflink>) Wider value of mathematics, e.g. <emph>Everybody will need to know some maths in their adult life</emph>.</p> <p>Further contextual instruments included a background questionnaire posted to all teachers midway through the school year, and covering basic demographic information, e.g. student gender, social class, ethnicity and language spoken at home (since non-fluent English might compromise an intervention grounded in dialogue). Teachers were asked to complete part from school records and to invite students to complete the remainder. Finally, teachers were invited periodically to report their practices and opinions about the materials. As detailed in Ruthven et al. ([<reflink idref="bib37" id="ref52">37</reflink>]), these reports confirmed broad similarity of lesson content across the intervention and control classes: there were no significant differences between the two groups over how they rated the suitability of knowledge test items for what had been covered. The reports also confirmed general compliance with the schedule for administering the evaluation instruments, with only four teachers (two intervention; two control) reporting delays. Opinion questionnaires, knowledge tests, attitude questionnaires, background questionnaires and teacher reports were returned by post after completion. Research assistants who were blind to whether instruments came from intervention or control classes coded and checked the data.</p> <hd id="AN0101854577-9">Preliminary analyses</hd> <p>Opinion questionnaires and full sets of knowledge tests were received from 11 intervention teachers (from nine different schools) and 16 control teachers (from 10 schools). The observational dataset comprised one lesson from each of seven teachers in the intervention group. All teachers returned attitude and background questionnaires, but some failed to ensure full completion of the latter. One consequence was inadequate responses to several indicators of social class, meaning that only crude 'eligibility for free school meals' was usable. Three teachers implemented the <emph>Ratios</emph> module with two classes, and in the interests of data independence only one of these classes was included in the analysis (the class in each pair that maximised intervention and control sample comparability over background characteristics).</p> <p>With the 25 constituent problems marked simply as correct or incorrect, Cronbach alphas for the knowledge tests were high (.85,.86, and.87 for the pre-test, immediate post-test and deferred post-test respectively). Accordingly, the tests were treated as single scales (score range = 0 to 25) for purposes of analysis, with two indices of knowledge gain computed for each student: (<reflink idref="bib1" id="ref53">1</reflink>) pre to immediate, i.e. immediate post-test score minus pre-test score; and (<reflink idref="bib2" id="ref54">2</reflink>) pre- to deferred, i.e. deferred post-test score minus pre-test score. Raw gain scores were used rather than residual gain scores because the latter adjust relatively weakly for associations between gain and pre-test. Stronger adjustment is possible through, e.g. ANCOVA on raw gains with pre-test as co-variate (Dimitrov & Rumrill, [<reflink idref="bib11" id="ref55">11</reflink>]).</p> <p>The Likert scales used in the opinion and attitude questionnaires were presented via qualitative options, i.e. <emph>Strongly agree</emph>, <emph>Agree</emph>, <emph>Tend to agree</emph>, <emph>Neither agree nor disagree</emph>, <emph>Tend to disagree</emph>, <emph>Disagree</emph> and <emph>Strongly disagree</emph>. For purposes of analysis, these options were transformed to values between +3 and -3. Scores for negatively worded items were reversed, so that positive scores always meant favourable opinions/attitudes. Factor analysis of data derived from the opinion questionnaire indicated that responses to 14 of the 20 items were loaded (.56 to.78) on a single factor, which accounted for 53% of the variance.[<reflink idref="bib3" id="ref56">3</reflink>] The remaining items were not strongly inter-correlated. Therefore, it was decided to base analyses on a single 14-item scale (Cronbach alpha =.93). A single-factor structure also emerged with the attitude data, this time accounting for 47% of the variance. All 20 items loaded on this factor (two at.45 but the remainder between.59 and.86), once more warranting treatment as a single scale (Cronbach alpha =.93). Analyses of opinion and attitude data were based on each student's average item scores of between +3 and -3.</p> <p>Two observational categories were discounted before analysis, "Different perspectives discussed for at least one minute" because it was never recorded and "Dialogic category involves at least three students" because it overlapped with other categories. The proportionate frequencies of the remaining categories were computed for each observed lesson (six minute periods in which the category occurred divided by total number of six minute periods), and factor analysis was used to map inter-category association. Two factors emerged with eigenvalues greater than one, and as Table 2 shows they account for substantial proportions of variance. Every category was loaded on a single factor, with loading strengths sufficient to indicate robustness despite derivation from only seven lessons (Guadagnoli & Velicer, [<reflink idref="bib14" id="ref57">14</reflink>]). One factor is interpretable as concerned with 'explanation', and as indicated in Table 2 it was detected in all classrooms, albeit to varying degrees. The other factor, interpretable as 'comparison', was more elusive occurring at best with very low frequency.</p> <p>Table 2. Factor analysis of observation category frequencies.</p> <p> <ephtml> <table><thead><tr><td /><td><bold>Comparison</bold></td><td><bold>Explanation</bold></td></tr></thead><tbody><tr><td><bold>Loadings for categories</bold></td></tr><tr><td> Teacher asks for explanation/clarification/reason</td><td char=".">−.33</td><td char="."><bold>.91</bold><sup>a</sup></td></tr><tr><td> Teacher puts student idea/question to whole class</td><td char="."><bold>.98</bold></td><td char=".">−.15</td></tr><tr><td> Teacher draws out difference between student ideas</td><td char="."><bold>.96</bold></td><td char=".">−.21</td></tr><tr><td> Teacher collects >1 student view without evaluating</td><td char=".">.<bold>96</bold></td><td char=".">.16</td></tr><tr><td> Teacher collects feedback from small group work</td><td char=".">.02</td><td char="."><bold>.68</bold></td></tr><tr><td> Student takes extended turn</td><td char=".">.35</td><td char="."><bold>.91</bold></td></tr><tr><td> Student gives reason</td><td char=".">−.33</td><td char="."><bold>.92</bold></td></tr><tr><td><bold>Factor parameters</bold></td></tr><tr><td> Variance explained</td><td>45%</td><td>44%</td></tr><tr><td> Mean frequency<sup>b</sup></td><td char=".">0.20</td><td char=".">1.52</td></tr><tr><td> Range<sup>b</sup></td><td char=".">0–0.90</td><td char=".">0.55–2.51</td></tr></tbody></table> </ephtml> </p> <p>1 Note: <sups>a</sups>Bold font in each column indicates the categories associated with the factor identified in the column heading; <sups>b</sups>From combined proportionate frequency of categories strongly loaded on each factor, so maxima are 3.00 for 'comparison' and 4.00 for 'explanation'.</p> <hd id="AN0101854577-10">Results</hd> <p>As regards knowledge gains and opinions, the key issue was benefits to the intervention students from following the <emph>Ratios</emph> module, to be addressed through comparison with the control students with background factors of possible relevance taken into account. Initially, the issue was addressed with each group considered separately, to see whether patterns differed. Although the interest was in student performance, the students were clustered amongst 27 school classes and in principle this could prove significant. Accordingly, the within-condition analyses employed robust standard errors that were clustered on the class variable, and were conducted using <emph>Stata Statistical Software</emph> (Version 12, StataCorp, College Station TX). Because clustering was found not to add precision in practice, the class-level variable was ignored when comparing across conditions. These comparisons were made using the <emph>Statistical Package for Social Sciences</emph> Version 21 (SPSS Inc., Chicago IL).</p> <p>Seven indicators of student background were deemed potentially relevant for knowledge gain with all but the seventh also potentially relevant for opinions: (<reflink idref="bib1" id="ref58">1</reflink>) gender; (<reflink idref="bib2" id="ref59">2</reflink>) social class; (<reflink idref="bib3" id="ref60">3</reflink>) ethnicity; (<reflink idref="bib4" id="ref61">4</reflink>) language used at home; (<reflink idref="bib5" id="ref62">5</reflink>) pre-test score; (<reflink idref="bib6" id="ref63">6</reflink>) score for the attitude questionnaire (Attitude score); and (<reflink idref="bib7" id="ref64">7</reflink>) score for the opinion questionnaire (Opinion score). A broad range of categories was used to assess ethnicity, but as only a handful of students selected any specific group apart from <emph>white</emph>, a simple <emph>white</emph> vs <emph>non-white</emph> dichotomy was used for purposes of analysis. Likewise, level of English language used at home was assessed as <emph>Always</emph>, <emph>Most of the time</emph>, <emph>Sometimes</emph>, <emph>Hardly ever</emph>, <emph>Never</emph>. However, as the final three options were seldom selected, a dichotomy was employed here too: <emph>predominantly</emph> (first two categories) vs <emph>other</emph> (remaining categories). Descriptive data for all background characteristics are presented in Table 3.</p> <p>Table 3. Student background characteristics.</p> <p> <ephtml> <table><thead><tr><td /><td>Intervention (<italic>I</italic>)</td><td>Control (<italic>C</italic>)</td><td /></tr><tr><td>Factor</td><td>Sample<sup>a</sup></td><td>Value</td><td>Sample</td><td>Value</td><td><italic>I</italic> vs <italic>C</italic> Comparison</td></tr></thead><tbody><tr><td>Gender<sup>b</sup></td><td char=".">272</td><td>50%</td><td>440</td><td>49%</td><td><italic>χ</italic><sup>2</sup>(1) = 0.01, <italic>p</italic> =.93</td></tr><tr><td>FSM eligibility<sup>c</sup></td><td char=".">240</td><td>12%</td><td>412</td><td>7%</td><td><italic>χ</italic><sup>2</sup>(1) = 5.13, <italic>p</italic> =.02</td></tr><tr><td>Ethnicity<sup>d</sup></td><td char=".">246</td><td>23%</td><td>408</td><td>13%</td><td><italic>χ</italic><sup>2</sup>(1) = 11.17, <italic>p</italic> =.001</td></tr><tr><td>Home language<sup>e</sup></td><td char=".">246</td><td>8%</td><td>411</td><td>1%</td><td><italic>χ</italic><sup>2</sup>(1) = 20.10, <italic>p</italic> <.001</td></tr><tr><td>Pre-test score<sup>f</sup></td><td char=".">278</td><td>12.64</td><td>440</td><td>13.60</td><td><italic>t</italic>(716) = 2.31, <italic>p</italic> =.02</td></tr><tr><td>Attitude score<sup>g</sup></td><td char=".">291</td><td>+1.17</td><td>442</td><td>+0.94</td><td><italic>t</italic>(731) = 2.96, <italic>p</italic> =.003</td></tr><tr><td>Opinion score<sup>g</sup></td><td char=".">236</td><td>+0.81</td><td>422</td><td>+0.80</td><td><italic>t</italic>(656) = 0.06, <italic>p</italic> =.95</td></tr></tbody></table> </ephtml> </p> <p>2 Note: <sups>a</sups> Numbers for whom data available, fluctuating due, e.g., to school absence, incomplete questionnaires; <sups>b</sups>Percentage of boys; <sups>c</sups>Percentage eligible for free school meals; <sups>d</sups>Percentage of non-white; <sups>e</sups>Percentage for whom English not main home language; <sups>f</sups>Mean out of 25; <sups>g</sups>Mean between +3 and -3.</p> <hd id="AN0101854577-11">Within-condition comparisons</hd> <p>Taking the intervention and control students separately and working with standardised scores, multiple regressions with clustered standard errors were used to examine which, if any, of the background characteristics predicted: (<reflink idref="bib1" id="ref65">1</reflink>) pre- to immediate knowledge gain; (<reflink idref="bib2" id="ref66">2</reflink>) pre- to deferred knowledge gain; and (<reflink idref="bib3" id="ref67">3</reflink>) opinion score. Analyses were conducted with all characteristics initially included, and repeated with characteristics systematically removed until 'best models' remained that contained only those predictors that achieved (or approximated) statistical significance. Presented in Table 4, these models in general explained relatively low proportions of variance. This was especially the case for knowledge gain with the intervention students where <emph>R</emph><sups>2</sups> was 9% for pre- to immediate gain and 7% for pre- to deferred gain. With knowledge gain amongst the control students and opinion scores amongst both groups, the explained proportions of variance were somewhat higher, but when <emph>R</emph><sups>2</sups> was between 14% and 24% still interpretable as modest.</p> <p>Table 4. Predictors of knowledge gain and opinion scores.</p> <p> <ephtml> <table><thead><tr><td /><td><bold>Beta</bold></td><td><bold><italic>t</italic>-value</bold></td><td><bold>Robust standard error (Non-clustered in brackets)</bold></td></tr></thead><tbody><tr><td><bold>Intervention</bold></td></tr><tr><td>Pre- to immediate gain (<italic>R</italic><sup>2</sup> =.09)</td></tr><tr><td> Pre-test score</td><td char=".">−.30</td><td char=".">−3.31, <italic>p</italic> =.008</td><td char=".">.09 (.06)</td></tr><tr><td>Pre- to deferred gain (<italic>R</italic><sup>2</sup> =.07)</td></tr><tr><td> Pre-test score</td><td char=".">−.27</td><td char=".">−2.75, <italic>p</italic> =.02</td><td char=".">.10 (.06)</td></tr><tr><td>Opinion score (<italic>R</italic><sup>2</sup> =.24)</td></tr><tr><td>Attitude score</td><td char=".">.45</td><td char=".">4.28, <italic>p</italic> =.003</td><td char=".">.11 (.07)</td></tr><tr><td>Ethnicity</td><td char=".">.17</td><td char=".">2.21, <italic>p</italic> =.06</td><td char=".">.08 (.16)</td></tr><tr><td><bold>Control</bold></td></tr><tr><td>Pre- to immediate gain (<italic>R</italic><sup>2</sup> =.19)</td></tr><tr><td> Pre-test score</td><td char=".">−.37</td><td char=".">−5.94, <italic>p</italic> <.001</td><td char=".">.06 (.05)</td></tr><tr><td> Gender</td><td char=".">.63</td><td char=".">6.44, <italic>p</italic> <.001</td><td char=".">.10 (.09)</td></tr><tr><td> Ethnicity</td><td char=".">.32</td><td char=".">2.72, <italic>p</italic> =.02</td><td char=".">.12 (.14)</td></tr><tr><td> Attitude score</td><td char=".">.15</td><td char=".">2.92, <italic>p</italic> =.01</td><td char=".">.05 (.05)</td></tr><tr><td>Pre- to deferred gain (<italic>R</italic><sup>2</sup> =.14)</td></tr><tr><td> Pre-test score</td><td char=".">−.37</td><td char=".">−8.44, <italic>p</italic> <.001</td><td char=".">.04 (.05)</td></tr><tr><td> Ethnicity</td><td char=".">.32</td><td char=".">2.50, <italic>p</italic> =.03</td><td char=".">.13 (.14)</td></tr><tr><td> Attitude score</td><td char=".">.23</td><td char=".">4.29, <italic>p</italic> =.001</td><td char=".">.05 (.05)</td></tr><tr><td>Opinion score (<italic>R</italic><sup>2</sup> =.20)</td></tr><tr><td> Attitude score</td><td char=".">.53</td><td char=".">10.59, <italic>p</italic> <.001</td><td char=".">.05 (.05)</td></tr><tr><td> Pre-test score</td><td char=".">−.19</td><td char=".">−3.39, <italic>p</italic> =.004</td><td char=".">.06 (.05)</td></tr></tbody></table> </ephtml> </p> <p>Nevertheless, despite their small contribution in absolute terms, pre-test scores significantly predicted both pre- to immediate gain and pre- to deferred gain with the intervention students. The negative coefficients indicate that students with relatively high pre-test scores progressed less than students with relatively low scores. As regards opinion scores, attitude scores and ethnicity (marginally) emerge as predictive amongst the intervention group. With positive coefficients in both cases (and <emph>non-white</emph> coded 1 and <emph>white</emph> coded 0), it appears that intervention students who held relatively positive attitudes to mathematics and/or were from minority ethnic backgrounds rated the <emph>Ratios</emph> module more positively than other students. Pre-test scores were also relevant for the two gain scores with the control students, operating in the same direction as for the intervention group. However gender, ethnicity and attitude scores were additionally associated with pre- to immediate gain, with the latter two factors also implicated in pre- to deferred gain. With positive coefficients in all cases, it seems that control students with relatively positive attitudes to mathematics and/or from minority ethnic backgrounds progressed the most. Moreover with <emph>male</emph> coded 1 and <emph>female</emph> coded 0, boys in the control group made more immediate progress than girls. As with the intervention students, opinion scores in the control group were positively predicted by attitudes to mathematics. This time however pre-test scores were also relevant, operating as negative predictors: control students who had relatively low pre-test scores rated their teaching the most favourably.</p> <hd id="AN0101854577-12">Between-condition comparisons</hd> <p>While the patterns that emerge in Table 4 are of intrinsic interest, they are also relevant for comparison across conditions. In particular, any factors that predict outcomes for both the intervention and control groups are potential confounds in between-condition comparison <emph>if</emph> they are also differentially distributed across the two conditions. Factors that are relevant with one condition only need not cause concern, nor need factors that apply with both conditions but whose distribution is equivalent. Judged against these criteria, pre-test score emerges as the sole potential confound with knowledge gain, negatively predicting pre- to immediate gain and pre- to deferred gain in both conditions, and as Table 3 shows, lower on average amongst the intervention students than the control group. Attitude score emerges as the only potential confound with opinion scores, positively associated with opinions in both the intervention and control conditions, and significantly higher amongst the intervention group.</p> <p>There are two possible responses to the potential confounds: to ignore them in the main between-condition analyses, or to incorporate them as covariates. Both are unsatisfactory, the former for obvious reasons and the latter because variance that is legitimately associated with the intervention vs control comparison will be removed from that comparison. This means that the magnitude of any between-condition differences will be under-estimated (Miller & Chapman, [<reflink idref="bib27" id="ref68">27</reflink>]). Recognising that neither strategy is above question, both were employed to compare the conditions in the hope that jointly they would provide interpretable results. As mentioned, these comparisons were no longer clustered for school class: as Table 4 shows, the standard errors obtained using non-clustered data are very similar to those obtained after clustering and certainly not the systematically smaller values that would exaggerate condition differences. Accordingly, t-tests were used to compare mean scores without taking account of potential confounds, and ANCOVA was used to make comparisons with potential confounds included.</p> <p>As shown in Table 5, the picture as regards knowledge gain is broadly the same regardless of analytic strategy. Mean pre- to immediate gain was higher in the intervention group than in the control group, but the differences were never statistically significant. Mean pre- to deferred gain was higher in both conditions than mean pre- to immediate gain, and the gap was greater in the intervention group than the control group. This resulted in consistently significant condition differences as regards pre- to deferred gain. The results obtained from the t-tests relating to opinions were presented in Table 3: they indicate virtually identical mean scores in the intervention and control groups. The condition differences remain non-significant after ANCOVA with attitude score as covariate, <emph>F</emph>(1600) = 2.38, <emph>p</emph> =.12<emph>.</emph> Estimated marginal means were +0.72 in the intervention group (<emph>SD</emph> = 0.94) and +0.84 in the control group (<emph>SD</emph> = 0.94).</p> <hd id="AN0101854577-13">Discussion</hd> <p>In many respects, the results are encouraging. As indicated in Table 5 (and no matter how the data were analysed), the intervention students progressed their understanding of proportional reasoning and rational number to a greater degree than the control students, with the differences statistically significant at deferred post-test. In addition, opinion scores amongst the <emph>epiSTEMe</emph> students averaged on the positive side of the scale (see Table 3). While these scores did not differ significantly from the control group, they nevertheless imply that the greater knowledge gains were not at the expense of opinions. Indeed data presented in Table 4 indicate little relation between knowledge and opinions in the intervention group: no attitudinal measure predicted knowledge gain nor vice versa. Within the control group by contrast, attitudes to mathematics were significant predictors of the two gain scores, and pre-test scores were significant predictors of opinions. This relative encapsulation amongst the <emph>epiSTEMe</emph> students partially explains why, as is clear from Table 4, background variables in general accounted for relatively low proportions of variance within this group. However, it is also relevant that responses from the <emph>epiSTEMe</emph> students were relatively independent of demographic factors. Opinion scores were related to ethnicity, but the relation fell short of statistical significance and contrasted with the much stronger relations involving both ethnicity and gender that were detected for the control group. Given concerns summarised earlier about student subgroups, this too can be seen as encouraging.</p> <p>Table 5. Mean knowledge gain (SD in brackets).</p> <p> <ephtml> <table><thead><tr><td /><td><bold>Intervention</bold></td><td><bold>Control</bold></td><td><bold>Comparison</bold></td></tr></thead><tbody><tr><td><bold><italic>t</italic>-test</bold></td></tr><tr><td>Pre- to immediate</td><td char=".">+0.97 (3.51)</td><td char=".">+0.65 (3.39)</td><td><italic>t</italic>(667) = 1.16, <italic>p</italic> =.25, <italic>d</italic> = 0.09<sup>a</sup></td></tr><tr><td>Pre- to deferred</td><td char=".">+1.67 (3.72)</td><td char=".">+0.85 (3.44)</td><td><italic>t</italic>(665) = 2.91, <italic>p</italic> =.004, <italic>d</italic> = 0.23</td></tr><tr><td><bold>ANCOVA (including pre-test)</bold></td></tr><tr><td>Pre- to immediate</td><td char=".">+0.84 (3.32)<sup>b</sup></td><td char=".">+0.73 (3.32)</td><td><italic>F</italic>(1,666) = 0.17, <italic>p</italic> =.68, <italic>d</italic> = 0.03</td></tr><tr><td>Pre- to deferred</td><td char=".">+1.53 (3.42)</td><td char=".">+0.94 (3.42)</td><td><italic>F</italic>(1,664) = 4.57, <italic>p</italic> =.03, <italic>d</italic> = 0.17</td></tr></tbody></table> </ephtml> </p> <p>3 Note: <sups>a</sups>Cohen's d is used to report effect sizes to facilitate comparison across analyses, and because standard objections to d ([<reflink idref="bib2" id="ref69">2</reflink>]) are unlikely to apply; <sups>b</sups>Means under ANCOVA are estimated marginal values.</p> <p>Nevertheless despite their positive features, the results obtained with the <emph>Ratios</emph> module require qualification. With Cohen's d for pre- to deferred gain lying between 0.17 and 0.23 (see Table 5), the benefits for student knowledge would conventionally be viewed as meaningful but rather small. Mean opinion scores are only slightly positive. Thus, two inter-related questions are raised: what was responsible for the observed benefits, and what could be done to boost effectiveness? As regards the first question, it seems unlikely that whole-class dialogue was playing a central role. For one thing, the observational data (see Table 2) indicate that dialogic teaching was not fully implemented: whilst the values for explanatory discourse mean that extended, reasoned contributions must have occurred (albeit not with consistently high frequency), the virtual absence of comparison implies that differences of perspective cannot have been dialogically resolved. In addition, given a major contribution from whole-class dialogue, variations in the frequency of key features would most likely have been associated with classroom averages for student outcomes. However, the similarity of standard errors obtained when the data were and were not clustered by classroom (Table 4) argues against substantial class-level effects, including by inference from whole-class dialogue.</p> <p>Of course, the module involved small group as well as whole-class activity, and the observational data were focused on the latter. It remains possible that dialogue occurring during group work did contribute to the positive results. Indeed, the possibility is supported through one specific finding, the growth in benefits between immediate and deferred post-tests. Growth is not merely indicated in Table 5, but from data presented earlier was also apparent during the informal evaluation at the end of the development period. Other research (e.g. Howe, [<reflink idref="bib17" id="ref70">17</reflink>]; Howe, McWilliam, & Cross, [<reflink idref="bib20" id="ref71">20</reflink>]) indicates that dialogue during group work frequently triggers delayed knowledge gain, and certainly this offers a more plausible interpretation of the progress between immediate and deferred post-tests than, say, teaching during the interval: the teacher reports mentioned earlier addressed coverage between the two post-tests, and without exception respondents said there had been no additional work. Nevertheless, even if small-group dialogue was important, it is unlikely to be the full story. All four <emph>epiSTEMe</emph> modules promoted dialogic practices at the whole-class and small-group levels, and all four obtained the whole-class patterns reported here, i.e. explanatory discourse to some degree but lesser use of comparison (see Ruthven et al., [<reflink idref="bib37" id="ref72">37</reflink>]). However, only one of the other modules was associated with knowledge gains that outstripped the control group, and here there was decrease between immediate and deferred post-tests (Howe et al., [<reflink idref="bib19" id="ref73">19</reflink>]).</p> <p>The implication is that if small-group dialogue contributed to the benefits associated with the <emph>Ratios</emph> module, it was in conjunction with the features that were relatively module-specific, i.e. the integrated treatment of proportional reasoning and rational number together with the consequent spotlighting of ratios. Importantly, this approach is not simply different from the UK norm; it is also somewhat discordant. As noted, the custom in the UK (as in many countries) is to ground teaching around fractions, whose coverage begins early in primary school. Yet as also noted, the ratio concept is widely regarded as simpler than other forms of rational number. Moreover, it links straightforwardly with sharing, e.g. three cakes between four children, and sharing is intuitively accessible from a very young age (Nunes & Bryant, [<reflink idref="bib30" id="ref74">30</reflink>]; Streefland, [<reflink idref="bib41" id="ref75">41</reflink>]). Furthermore, while proportional reasoning can be challenging (Lamon, [<reflink idref="bib23" id="ref76">23</reflink>]; Lesh, Post, & Behr, [<reflink idref="bib24" id="ref77">24</reflink>]; Piaget, Grize, Szeminska, & Bang, [<reflink idref="bib31" id="ref78">31</reflink>]), even six-year-olds can achieve basic understanding with support (Carraher, [<reflink idref="bib6" id="ref79">6</reflink>]). Thus, moving to the second question highlighted above, perhaps one way to capitalise on the <emph>Ratios</emph> module would be to introduce the approach before Year 7, possibly even reversing the traditional sequence by starting with ratios and leaving fractions until later. In this context, it seems unfortunate that recent policy within the UK confirms the initial focus on fractions, while mandating introduction at an even earlier stage in the primary curriculum (DfE, [<reflink idref="bib10" id="ref80">10</reflink>]).</p> <p>Given the uncertainties discussed already, it is unclear whether the emphasis as regards enhancement should merely be upon the module-specific principles, or whether dialogue should also be addressed. Nevertheless, if future research endorses the relevance of small group dialogue as suggested above (and especially if endorsement also applies at the whole-class level), further action would be required here too, particularly over the practices characterised as comparison. The reason why comparison proved more challenging than explanation probably lies with its greater divergence from the initiation-response-feedback sequences that traditionally dominate classroom interaction (Howe & Abedin, [<reflink idref="bib18" id="ref81">18</reflink>]): while it should not be overly difficult to follow "What fraction is orange [initiation]?" "One tenth [response]" with "Can you tell everyone how you worked that out [trigger to explanation]?" rather than "Very good [feedback]", comparison requires greater adjustment. Whatever the case though, it is clear that the two days of professional development associated with the formal evaluation were insufficient to realise <emph>epiSTEMe's</emph> aims relating to dialogue. Yet as noted, two days are the norm for educational innovation within the UK, implying significant consequences for policy should dialogue genuinely need to be addressed. Further research is required here, and perhaps the main point to make in conclusion that given the promise that the <emph>Ratios</emph> module already exemplifies such research is well worth conducting.</p> <ref id="AN0101854577-14"> <title> Notes </title> <blist> <bibl id="bib1" idref="ref18" type="bt">1</bibl> <bibtext> At the time of the study, around 80% of students achieved at least this level. Thus, while lower achieving students (and lower ability sets) will have been excluded, a wide spectrum will have been sampled with no constraints at the upper end.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref19" type="bt">2</bibl> <bibtext> In England, CVA2-4 is one of several nationwide and standardised indices of efficacy. 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  Data: Rational Number and Proportional Reasoning in Early Secondary School: Towards Principled Improvement in Mathematics
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  Data: <searchLink fieldCode="AR" term="%22Howe%2C+Christine%22">Howe, Christine</searchLink><br /><searchLink fieldCode="AR" term="%22Luthman%2C+Stefanie%22">Luthman, Stefanie</searchLink><br /><searchLink fieldCode="AR" term="%22Ruthven%2C+Kenneth%22">Ruthven, Kenneth</searchLink><br /><searchLink fieldCode="AR" term="%22Mercer%2C+Neil%22">Mercer, Neil</searchLink><br /><searchLink fieldCode="AR" term="%22Hofmann%2C+Riikka%22">Hofmann, Riikka</searchLink><br /><searchLink fieldCode="AR" term="%22Ilie%2C+Sonia%22">Ilie, Sonia</searchLink><br /><searchLink fieldCode="AR" term="%22Guardia%2C+Paula%22">Guardia, Paula</searchLink>
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  Data: 10.1080/14794802.2015.1019914
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  Data: Reflecting concerns about student attainment and participation in mathematics and science, the Effecting Principled Improvement in STEM Education ("epiSTEMe") project attempted to support pedagogical advancement in these two disciplines. Using principles identified as effective in the research literature (and combining these in a novel fashion), the project developed topic modules for early secondary-school teaching in the UK, arranged for their implementation in classrooms, and evaluated the results. This article reports the development, implementation and evaluation of the "epiSTEMe" mathematics module entitled "Fractions, Ratios and Proportions." The module covers aspects of rational number and proportional reasoning relevant to the early secondary curriculum, and was developed in collaboration with teachers, implemented in 11 classrooms, and evaluated through comparison with 16 control classrooms where the topic was addressed using established methods. Students who used the "epiSTEMe" materials made significantly greater progress than control students as regards topic mastery, while holding positive opinions about their teaching and learning experiences.
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