Sequencing computer-assisted learning of

Teaching Mathematics and Its Applications (2011) 30, 120^137 doi:10.1093/teamat/hrr009

Advance Access publication 9 June 2011

Sequencing computer-assisted learning of transformations of trigonometric functions JOHN A.ROSS,CATHERINE D.BRUCE` AND TIMOTHY M. SIBBALD‰ †

University of Toronto, OISE/Trent Valley Centre, PO Box 7190, 1994 Fisher Drive, Peterborough, ON K9J 7A1, Canada, ` School of Education & Professional Learning, Trent University, 1600 West Bank Drive, Peterborough, ON K9J 7B8, Canada, § Thames Valley District School Board, 34 Stonehenge Pl, London, ON N5V 4C5, Canada

Email: [email protected]

[Submitted December 2010; accepted March 2011]

Studies incorporating technology into the teaching of trigonometry, although sparse, have demonstrated positive effects on student achievement. The optimal sequence for integrating technology with teacher-led mathematics instruction has not been determined. Our research investigated whether technology has a greater impact on student achievement and attitudes if it is implemented before or after whole class teaching. The curriculum context of the study was a set of learning objects (CLIPS: Trig) designed to support student learning of transformations of trigonometric functions. The software includes functional features identified in prior research: it relieves students of the tedium of creating graphs by hand; sliders give students control of the simulations within program parameters; there are easy transitions between algebraic and graphic representations; the environment is dynamic; animation and visualization are included with graphing functions. Twenty Canadian classrooms (N = 489 grade 11^12 students, aged 17^18 years) were randomly assigned to two instructional sequences: CLIPS: Trig followed by whole-class teaching (CLIPS early treatment) and whole-class teaching followed by CLIPS: Trig (CLIPS late treatment). We found that in the pre-test to post-test comparisons, students who experienced CLIPS: Trig after whole-class teaching of core concepts learned more than students who began the unit with technology-supported simulations. However, there were no statistically significant differences in the pre-test to delayed post-comparisons. Beginning the trigonometry unit with CLIPS: Trig enhanced the impact of whole-class teaching, while beginning with whole-class teaching enriched students’ technology experience. The findings suggest that a tight integration of whole-class and technology-assisted instruction is preferable.

1. Introduction Research on the teaching and learning of trigonometry, with and without technological aids, lags research conducted in other domains of mathematics education. As Steckroth (2007) noted in her necessarily brief review, ‘the research dealing with trigonometric functions is sparse, especially studies which concentrate on student achievement in trigonometry’ (p. 51). We begin this article by reminding ß The Author 2011. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: [email protected]

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readers of the importance of trigonometry in the secondary school curriculum, identify key findings from the limited research that has been reported, and describe the results of an investigation that compared the student achievement effects of two approaches to integrating an interactive software program with whole-class instruction. The research, conducted on a set of interactive learning objects developed for Canadian students in grades 11 and 12 (17–18-year old), found that the technology was more effective if delivered after students had been introduced to core trigonometric concepts through whole-class teaching. However, by the end of the study when students had experienced whole class and technology-assisted instruction there were no statistically significant differences between the sequences.

2. Research on technology-aided instruction of trigonometric concepts 2.1

The rationale for investigating the teaching of trigonometry

Trigonometry is a core part of the secondary school curriculum. In the United States, the Standards of the National Council of Teachers of Mathematics (NCTM, 2000) highlight the importance of trigonometry in the study of functions, particularly periodic functions, and emphasize trigonometry’s utility in investigating real-world phenomena. Trigonometric understanding is prerequisite to calculus and has applications in Newtonian physics, astronomy, architecture, surveying and engineering (Steckroth, 2007). In England, tasks involving trigonometry made up 32% of items on the A-level mathematics examinations of 1999–2001 (Delice & Roper, 2006) and trigonometry topics have a prominent place in the curriculum priorities of such disparate nations as Australia (Kendall & Stacey, 1996), Canada (Colgan, 1992), Korea (Choi-Koh, 2003), Singapore (Ng & Hu, 2006) and Turkey (Bintas & Sarsar, 2009). Trigonometry is part of the advanced mathematics curriculum of all 48 countries sampled by TIMSS (Trends in International Mathematics and Science Study) in 2007, although there is variation in the amount of emphasis given to the domain (Garden et al., 2008). The use of trigonometric transformational geometry is relatively recent in many countries which may explain the relatively sparse literature in the area. Students find trigonometry challenging and it has been described as the hardest part of the secondary school mathematics curriculum (Takaˇci et al., 2005). Deep understanding of trigonometry requires the ability to flip between abstract, visual and concrete representations of mathematical objects. In addition, the subject is confounded by inter-relationships between functions (e.g. a cosine is a sine that has been horizontally shifted). Students are particularly handicapped by their inability to formulate and transpose algebraic expressions and by misconceptions about core constructs such as function. Many students rely on memorization (using mnemonics such as SOH-CAH-TOA, meaning Sine is the ratio of the Opposite side to the Hypotenuse and so on), with mechanical recital of algorithms through superficially similar problems. Although all students in the sample of Kendall and Stacey (1996) had studied trigonometry a year prior to enrollment in their study, most scored zero on an assessment of trigonometric knowledge. Even the high-school students who correctly visualized the counter-intuitive location of a horizontal transformation in Zaskis et al. (2003) were unable to give a credible mathematical explanation for their prediction. The undergraduates in Weber’s (2005) sample were unable to draw from their high-school training conceptual explanations of their trigonometric reasoning prior to participating in an innovative program.

2.2

Technology-assisted learning devices

Some of the sources of learning challenge can be addressed with assistive technology. Graphing calculators and software can relieve students (and their teachers) of the tedium of sketching graphs

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TRANSFORMATIONS OF TRIGONOMETRIC FUNCTIONS

by hand, freeing mental energy for conceptual development (Kissane & Kemp, 2009). Choi-Koh (2003) found that the zoom-in feature of some calculators enabled students to focus on particular sections of a graph, while the entire graph was visible on a split screen. Many calculators facilitate easy translation from graphs to algebraic representation of functions, contributing to the development of students’ ‘function sense’; i.e. the intersection of number, symbol and graph sense (Kakihana et al., 2003). Using technology to support trigonometric learning enhances student attitudes to mathematics learning (Choi-Koh, 2003). Other technology-related features not specifically investigated by trigonometry researchers may also contribute to student learning. Giving students access to Computer Algebra Systems may compensate for deficiencies in solving algebraic tasks embedded in trigonometry problems. Research in other mathematics domains demonstrates that use of Computer Algebra Systems increases student attention on notation, on relationships, and on the meaning of symbols, extending the complexity of algebraic expressions that students can handle which frees them to focus on conceptual issues without loss of computational skill (Heid, 1997). Software with interactive features may increase students’ sense of control over their learning (Ainley, 2000), increasing student willingness to engage in trigonometry tasks. Inclusion of dynamic features enables the software to illustrate mathematical changes that might not be otherwise visible and helps students visualize a dynamic model containing trigonometric relationships that are difficult to depict with static images (Ng & Hu, 2006). Studies incorporating technology into the teaching of trigonometry have demonstrated largely positive effects on student achievement (Blackett & Tall, 1991; Choi-Koh, 2003; Ng & Hu, 2006; Bintas & Sarsa, 2009) with one study finding no impact (Colgan, 1992) and another reporting mixed results: Delice and Roper (2006) found that Turkish students who did not have access to technology outperformed English students who did on the algebraic and manipulative aspects of trigonometry. But the English- and Turkish-learning experiences differed in many other ways and there was no evidence presented about the school participation rates of 16–18-year olds in the two countries. In addition, the English students outperformed the Turkish on applications of trigonometry to practical problems. There are added benefits when the technology includes more dynamic learning opportunities. For example, Steckroth (2007) found that software that included animation and visualization produced greater learning than software limited to graphing functions. Although research on learning trigonometry is bedeviled by small samples, non-comparative designs and the confounding effects of multiple treatment elements, it provides reasonable grounds for anticipating that the inclusion of technology will benefit students. A strong conclusion from this research is that giving students access to technology is not sufficient to bring about learning, a finding that is shared with inquiries into technology-based teaching in other domains of mathematics learning (Heid & Blume, 2008). Students may be distracted by visual cues, focusing on surface features of practical tasks, rather than on trigonometric concepts. Prompting by the teacher (as in the one-to-one tutoring of Choi-Koh, 2003), whole class discussions and/or cooperative learning to stimulate rich mathematical discourse among students (Ng & Hu, 2006; Bintas & Sarsar, 2009) are recurrent themes in trigonometry success stories.

2.3

Gaps in prior research

Several applets or learning objects [some are reviewed by Kissane & Kemp (2009)] have been developed to address particular trigonometry learning outcomes. They are intended to be used in conjunction with whole-class instruction. The learning objects for teaching transformations of trigonometric functions that we investigated were of this type. An important unresolved issue in using such devices is sequencing. Is it better to begin the instructional sequence with the learning

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objects to develop core concepts or is it better to begin with whole-class instruction and use the learning objects to consolidate student understanding? Choi-Koh (2003) concluded that ‘if students use tools early in the learning environment to acquire a broader picture of the ways in which concepts may be realized, they might progress cognitively’ (p. 367) but Choi-Koh’s study investigated only a single condition—technology introduced after the students had a year of trigonometry instruction, which might be construed as support for late introduction. Lesser and Tchoshanov (2005) presented evidence that students need to be taught abstract, visual and concrete representations to develop ‘function sense’ (the ability to integrate and flexibly apply multiple representations of functions). They found that the optimal sequence for introducing representations in trigonometry is to present the abstract first; the visual and concrete became meaningful only after the abstract had been learned. Several themes from other domains of mathematics learning have not been part of research on trigonometry education. For example, student attitudes affect the impact of technology, especially student perceptions of their ability to solve mathematical problems and attitudes to using technology to learn mathematics (Moos & Azevedo, 2009). Student attitudes have been rarely investigated as an outcome of technology-supported trigonometry instruction [an exception is Colgan (1992) who found null effects]. Only one study distinguished the effects of technology on learning trigonometry ratios by important student subgroups. Blackett and Tall (1991) found that females benefitted more than males from graphic calculators in a study of trigonometric achievement. No study has decomposed its results by ability group.

3. Research questions Our study was guided by research questions derived from our review of previous research on technology-based approaches to trigonometry teaching. Does the sequence of technology implementation (before or after whole-class teaching) have an impact on student achievement and/or student attitudes? If sequencing does make a difference to student outcomes, are the effects influenced by student characteristics such as gender and ability?

4. Methodology 4.1

Curriculum context

The focus of our research was ‘CLIPS: Trig’, a set of learning objects developed by the Ontario (Canada) Ministry of Education to support student understanding of transformations of trigonometric functions. CLIPS (Critical Learning Instructional Paths Supports) are multi-media learning objects focused on important curriculum expectations. The term CLIPS was coined by Hill and Cre´vola as ‘devices for bringing expert knowledge to bear on the detailed daily decisions that every teacher must make in teaching a coherent domain of the curriculum’ (Fullan et al., 2006, p. 56). The Ministry has developed several sets of CLIPS for challenging mathematical topics at various grade levels (www .mathclips.ca). Previous research found that CLIPS: Fractions developed for students in grades 7–10 (13–16-year old) contributed to student achievement (Ross & Bruce, 2009) when appropriately implemented—for example, when hardware requirements are met, low-achieving students are selected for the program and prepared with prerequisite knowledge, and teachers debrief students during and after CLIPS activities (Bruce & Ross, 2009; Ross et al., 2009). In the fractions studies, CLIPS: Fractions were introduced following whole-class instruction. In the CLIPS: Algebra study (not yet completed), CLIPS were integrated with whole-class instruction. The current study is the only investigation I the CLIPS series to systematically vary sequencing.

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CLIPS:Trig addresses five curricular objectives: (i) defining trigonometric functions by means of practical examples such as swimming in a pool, tides and circular motion; (ii) defining constructs and showing how the defined items are affected by changes in graphs; (iii) establishing the graphing of the sine function and how the defined properties are determined from the context of the graphing; (iv) transformations of the graphs and how the parameters of the algebraic model change as well as the properties of the graphs; (v) applying the properties and transformations to practical examples. The five objectives are sequenced through seven interactive sets of activities involving approximately 150 min of classroom time. The software includes formative feedback including quizzes. CLIPS: Trig includes many of the functional features identified in prior research: it relieves students of the tedium of creating graphs by hand (Kissane & Kemp, 2009); sliders give students control of the simulations within program parameters (Ainley, 2000); there is easy transition among representations (Kakihana et al., 2003); the environment is dynamic (Ng & Hu, 2006); animation and visualization are included with graphing functions (Steckroth, 2007). Teacher preparation for CLIPS: Trig implementation includes guidelines drawn from research on CLIPS: Fractions; for example, ensure hardware requirements are met (e.g. Flash capability); intervene during student work if students become confused; have students work through the program in pairs to encourage mathematical talk (Bruce & Ross, 2009). Both treatments used the unit circle method and some review of the ratio method (which is taught a year earlier) as described in Kendall & Stacey (1996). Whole-class instruction covered the same trigonometric concepts as CLIPS: Trig but in a less dynamic way. For example, having a circle going around and the sine curve rolling off (as in CLIPS: Trig) is almost impossible to do without the aid of technology. Teachers typically used pairs of overheads, one with a graph and the other with a sine curve, to show the movement of the curve relative to the graph. With a few exceptions, real-world connections were given cursory mention. In contrast with CLIPS: Trig, definitions were less compartmentalized. Classroom instruction typically entailed more graphing by hand. Teachers emphasized algebraic formulas and their connection between algebraic and graphical representations, as opposed to the multiple representations (concrete, algebraic, abstract and graphical) available in CLIPS: Trig.

4.2

Research design

The study was a randomized field trial in which the sequence of CLIPS use (before or after whole-class instruction) was investigated as a predictor of student achievement. Table 1 shows there were three test occasions. In the first we measured student performance on prerequisite knowledge and attitudes on TABLE 1. Research design Prior assessment

First treatment

Interim assessment

Second treatment

Final assessment

O1abc O1abc

XCLIPS Xwhole class

O2abc O2ab

Xwhole class XCLIPS

O3 O3

O1a = Assessment of student prerequisite knowledge. O1b = Assessment of student attitudes. O1c = Assessment of teacher readiness to use technology in math teaching. O2a = Assessment of student achievement. O2b = Assessment of student attitudes [O1b and O2b are the same measures]. O2c = Assessment of teacher implementation of CLIPS. O3 = Final assessment of student achievement. XCLIPS = Instruction delivered through CLIPS: Trig. Xwhole class = Instruction on Trigonmetric functions delivered through delivered through whole class teaching.

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entry to the study. The second, following instruction in transformations delivered through CLIPS or whole-class teaching, measured student achievement and attitudes. The third, following a reversal of conditions so that students who had received CLIPS now received whole-class teaching and vice versa, measured student achievement. The first row in the table represents the experience in the early CLIPS condition; the second row represents students in the late CLIPS condition.

4.3

Sample

Math coordinators in two school districts invited teachers of grades 11 and 12 courses that include Trigonometric Transformations as curriculum expectations to participate in the study. Twenty teachers (N = 537 students, reduced to 458 as described below) were randomly assigned within districts to two study conditions: CLIPS early and CLIPS late. All teachers were certified to teach secondary school mathematics and virtually all had majored in mathematics or in a cognate subject such as statistics or engineering. The majority had participated in summer institutes on mathematics education (60%) and attended mathematics conferences (85%). A minority (40%) had taken additional qualification courses in mathematics and two of the 20 teachers had participated in graduate studies. The teachers varied in career experience; they were evenly distributed from 20 years as mathematics teachers.

4.4

Student instruments

Student outcomes consisted of achievement measures (multiple-choice and open-ended), beliefs about self (mathematics self-efficacy, technology self-efficacy and confidence in Trigonometry knowledge), beliefs about mathematics learning (functional or dysfunctional) and self-reported effort in mathematics class. There were four student moderators: gender, grade, stream (4-year-university or 2-year-college bound) and whether the student had taken the course before. The assessments of student achievement each contained 20 multiple-choice items, marked correct/ incorrect. There were also one open-ended item on the pre-test and two open-ended items on the post-test and final test. Items for achievement tests 2 and 3 (post-test and final test) were developed in pairs and randomly assigned to the two test forms. Items were field tested with a small sample of grade 11 students and their teachers to obtain evidence of face validity. The open-ended items were scored 0–4 based on criterion referenced rubrics. There was perfect agreement between the marker and the trainer on 88% of a sample of 60 items, = 0.85. The achievement tests generated two measures each: The mean score on the multiple-choice and the mean score on the open-ended questions. Mathematics self-efficacy consisted of the mean score on six items from Ross et al. (2002); e.g. ‘as you work through a math problem how sure are you that you can. . . explain the solution’). Technology self-efficacy consisted of the mean on five items adapted from Pierce et al. (2007), for example, ‘I can learn any computer program needed for school’. Confidence in domain-specific knowledge consisted of the mean on six items developed for this study that measure how confident students are in their Trigonometric knowledge, such as ‘I will have trouble relating algebra of lines and/or quadratics to their graphs.’ Functional beliefs about technology & mathematics learning consisted of the mean on five items adopted from Pierce et al. (2007), such as ‘computers help me learn mathematics better’. Quick/fixed learning or dysfunctional beliefs about mathematics learning consisted of the mean of eight items adapted from Schommer-Aitkins et al. (2005). These items measure belief in quick/fixed learning (i.e. that learning occurs quickly or not at all and that intelligence is fixed rather than incremental); e.g. ‘If I cannot understand something quickly,

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it usually means I will never understand it.’ Effort in mathematics class consisted of the mean of 11 items developed for this study, for example, ‘most of the time . . . how hard do you concentrate in math class?’ Student background consisted of gender, grade (11–12), stream and whether the student had taken the course before.

4.5

Teacher instruments

Although the focus of the study was student outcomes, we included teacher variables to measure the extent to which teachers in the two conditions were equivalent. Mathematics teaching practices was the mean of 20 items measuring teachers’ self-reported implementation of NCTM Standards from Ross et al. (2003). Personal teaching efficacy consists of three scales adapted for mathematics from Tschannen-Moran & Wolfolk Hoy (2001). Efficacy for engagement, efficacy for teaching strategies and efficacy for student management were each the mean of four items. Use of technology in mathematics class was the mean of 10 items such as ‘how often do you . . . use a computer as a presentation tool?’ Items were adapted from Drent and Meelissen (2008) and Mueller et al. (2008). Christensen et al. (2005) provided eight items measuring teacher Concern about Technology (e.g. ‘Computers are changing the world too rapidly’) and five items for Significance of technology (e.g. ‘Having computer skills helps one get better jobs’). Background in teaching mathematics consisted of four items probing certification for teaching mathematics, undergraduate training in mathematics, professional learning experiences in mathematics and career teaching experience. At the end of the study, teachers completed an implementation survey that recorded time spent on CLIPS: Trig, strategies for integrating technology with whole class teaching, problems encountered and solutions applied.

4.6

Qualitative data

The qualitative methods involved a multiple instrument case study design (Yin, 2009) in which we collected descriptive data of participant activity. We observed teachers and students at two case sites, taking field notes using a consistent observation guide. For the early CLIPS case study, we observed two periods of CLIPS: Trig use as well as the first class after CLIPS: Trig use, in order to observe the transition back to regular classroom instruction, and to record further student responses to CLIPS: Trig. For the late CLIPS group, we attended the day prior to CLIPS: Trig implementation to observe the transition from regular instruction to CLIPS: Trig, followed by observations of CLIPS: Trig activity, and the day following CLIPS: Trig implementation to observe the transition back to regular instruction. We sat beside students and asked probing questions as they navigated through the online learning environment. We took notes and audio recorded student activity and dialogue. Following CLIPS: Trig use, we conducted interviews with the two case study teachers and their students. We also conducted individual and focus group interviews with teachers in both districts. The qualitative data consisted of 12 sets of field notes (nine classroom observations and notes on in-service activity), 13 individual student interviews (121 min of audio), three student focus groups (64 min of audio), six individual teacher interviews (148 min of audio) and four teacher focus groups (45 min of audio as well as field notes).

4.7

Data analysis

We began the quantitative analysis by recoding items, searching for missing values and outliers, creating scales, examining distributional properties and determining reliability. We followed Slavin

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(2008) who recommended randomized quasi-experimental designs in which students are randomly assigned in classes but data are analyzed at the individual level. This procedure is suitable when the equivalence of the student groups has been demonstrated. We answered the first research question, ‘To what extent does CLIPS: Trig contribute to student achievement and affective outcomes?’ by using within-subjects analysis (GLM in SPSS/PASW 18); the repeated measures were pre- and post-test scores on student outcomes. In subsequent analyses, the repeated measures were post-test and delayed post-final test scores and then pre-test and delayed post-final comparisons. The minimum detectable effect size for a within-subjects design with two experimental groups for a sample of this size is 0.14, at 80% power, p = 0.05 (Dennis, 1994). To address the second research question ‘Are the effects of CLIPS use moderated by students’ capacity for mathematics learning?’, we repeated the withinsubjects analysis with potential moderators as between-subjects factors. We conducted an inductive analysis of codes generated from interview transcripts and observation field notes (Bogdan & Biklen, 2003). We searched for themes that were common to all subjects and which emerged from common practices and perspectives shared among the participants (Creswell, 2007), namely attitudes and beliefs about the shared experience of learning from CLIPS. We identified themes and start codes through transcript and field-note analysis, taking anomalies and contradictions into consideration. Code counts were conducted to ensure that themes were robust in frequency and power.

5. Results 5.1

Descriptive analysis

Some students (N = 79) completed the pre- but not the post-test survey. Students were less likely to complete both surveys if they were in the early group that experienced CLIPS: Trig before whole class teaching than if they were in the late treatment that experienced CLIPS: Trig after whole-class teaching: 18% were missing the post-test survey in the early condition and 10% were missing the post-test survey in the late condition (2 = 5.90, df = 1, p = 0.018). However, there were no statistically significant differences between missing and not missing groups on any of the pre-test variables (achievement, attitude and student demographics). We deleted students who had not completed both surveys, reducing the sample to 458 students. We also investigated whether students in both conditions were taught by teachers with similar characteristics. We found there were no statistically significant differences on any of the teacher measures (self-reported teaching practices, three measures of teacher efficacy and three measures of attitudes to technology). These comparisons are not tabled. Completion of the third achievement test was presented to teachers as an optional activity. The students who completed the third test were not representative of the total sample. Students in the late treatment (62%) were more likely than students in the early treatment (41%) to complete the third test (2 = 20.95, df = 1, p = 0.001). Students who completed the third test scored significantly lower on most pre-test variables (mathematics self-efficacy, trigonometry self-efficacy, functional attitudes to technology and achievement) and lower on post-test measures of mathematics self-efficacy and trigonometry self-efficacy. There were no other differences, including for the dichotomous variables (gender, grade and repeating the course). After deleting students who did not complete both surveys, we replaced missing values (