Academic music: music instruction to engage third-grade students in learning basic fraction concepts
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DOI: 10.1007/s10649-012-9395-9
- Cite this article as:
- Courey, S.J., Balogh, E., Siker, J.R. et al. Educ Stud Math (2012) 81: 251. doi:10.1007/s10649-012-9395-9
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Abstract
This study examined the effects of an academic music intervention on conceptual understanding of music notation, fraction symbols, fraction size, and equivalency of third graders from a multicultural, mixed socio-economic public school setting. Students (N = 67) were assigned by class to their general education mathematics program or to receive academic music instruction two times/week, 45 min/session, for 6 weeks. Academic music students used their conceptual understanding of music and fraction concepts to inform their solutions to fraction computation problems. Linear regression and t tests revealed statistically significant differences between experimental and comparison students’ music and fraction concepts, and fraction computation at posttest with large effect sizes. Students who came to instruction with less fraction knowledge responded well to instruction and produced posttest scores similar to their higher achieving peers.
Keywords
Fraction conceptsElementaryRepresentationMusic notationSemiotics1 Introduction
Fractions are one of the most difficult mathematical concepts to master in the elementary curriculum (Behr, Wachsmuth, Post & Lesh, 1984; Cramer, Post & delMas, 2002; Moss & Case, 1999). For many students, the struggle to understand fractions continues through middle and high school, thereby delaying or preventing development of mathematical reasoning and mastery of algebraic concepts (Brigham, Wilson, Jones & Moisio, 1996; Mazzocco & Devlin, 2008; National Research Council, 2001). For students with learning difficulties, English Language Learners (ELLs), and students with low achievement, mastering fraction concepts is an even more formidable task (Basurto, 1999; Butler, Miller, Crehan, Babbitt & Pierce, 2003; Empson, 2003; Hiebert, Wearne & Taber, 1991; Mazzocco & Devlin, 2008; Menken, 2006).
Fraction instruction, like other content areas in the elementary curriculum, has undergone two large changes brought on by the school reform movement. First, in response to past poor performance of US students on international mathematics assessments (McLaughlin, Shepard & O’Day, 1995), the National Council of Teachers of Mathematics (NCTM) set forth specific standards for mathematical reform (Maccini & Gagnon, 2002; NCTM, 2000). Included in the NCTM standards are specific goals and recommendations for curricular changes regarding fractions and fraction instruction. More recently, many states are adopting the Common Core standards, in which developing an understanding of fraction equivalence, reasoning about size, and understanding a fraction as a number on a number line become even more important parts of the third-grade math curriculum. Second, with the implementation of No Child Left Behind of 2001, its impending reauthorization, and the reauthorization of the Individuals with Disabilities Education Act (IDEA) in 2004, effective mathematics achievement for all students within the least restrictive environment must be realized and all students must be prepared to meet proficiency goals on statewide assessments.
NCTM and Common Core standards related to fractions for elementary through middle school emphasize conceptual understanding. In turn, reform-based fraction instruction has shifted from more behavioral, procedurally driven, rule-based approaches to a more cognitive conceptual approach with emphasis on problem solving and reasoning (Butler et al., 2003; Woodward, Baxter & Robinson, 1999). Fraction size includes knowing a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b parts and understanding a fraction a/b as the quantity formed by a parts of size 1/b. Understanding fraction size also includes identifying a fraction on a number line. Equivalency refers to different fractions representing the same rational number. NCTM (2000) standards suggest that young children be exposed to simple fractions through real-life experiences and language. These young students should understand and be able to visually represent commonly used fractions such as one half, one fourth, and one eighth. In grades three to five, students should (a) develop an understanding of fractions as parts of wholes, as parts of a collection, as locations on the number line, and as divisions of whole numbers, (b) use models, benchmarks, and equivalent forms to judge the size of fractions, and (c) recognize and generate equivalent forms of commonly used fractions. The Common Core fraction standards for third grade are similar but they also include placing and identifying fractions on a number line.
According to these standards, students should to be able to convert a number expressed in terms of one unit to another unit. To understand this, students must realize that the magnitude of the wholes for equivalent fractions should be the same (Post, Behr & Lesh, 1986). Two fractions can only be compared when the sizes of the wholes are equal (Yoshida & Sawano, 2002). Moving between and manipulating fractions, with the underlying concept of the invariance of the whole, helps students to understand the order of fractions as quantities rather than as symbols. A fragile understanding of fractional parts is the barrier that prevents students from mastering computation skills with fractional quantities (Young-Loveridge, Taylor, Hawera & Sharma, 2007). Yoshida and Shinmachi (1999) suggest expressly teaching the equal-whole schema for students to more easily understand the concepts of equivalence and magnitude. In the present study, we have designed a fraction instructional intervention that utilizes the proportional representation of fractional quantities inherent in musical notes.
Techniques and strategies to teach difficult introductory fraction concepts to groups of heterogeneous students in general education classrooms should be designed with efficacy and ease of implementation in mind. An instructional strategy must address students’ difficulty in shifting from whole to rational number reasoning with a focus on developing understanding of fractions as numbers (Mack, 1995). The instructional design must also address teachers’ discomfort with teaching difficult mathematics concepts (Frykholm, 2004) and the lack of conceptual instructional guidance in classroom materials and texts (Sood & Jitendra, 2007). Considering teachers’ comfort level with teaching difficult content is especially important because each lesson has to be designed so that a teacher without music instruction experience or training could deliver the intervention with fidelity. We were careful to include clear directions and worked examples with each lesson plan and worksheet. In order for our intervention to have social validity (Foster & Mash, 1999), we wanted to be sure that teachers could use Academic Music without the help of a music teacher. In addition, we sought to make each lesson fun and entertaining for both the teacher and the students.
2 Theoretical perspective: a semiotic approach to instruction
2.1 Semiotics
The heart of mathematical development is symbolic representation of an actual quantity. This symbolic representation consists of signs and rules that bear intentional characteristics and goals. However, novice learners may not readily perceive the meanings behind symbolic representation. Thus, from an educational stance, the process behind how young children construct symbolic meanings becomes an important question. Influenced largely by Vygotsky’s work (see Vygotsky, 1978), mathematical development has been viewed as learning tools and signs that are elements of specific communicative systems, which vary by culture. According to semiotic perspectives, symbolizing involves manipulations with cultural tools such as physical objects, pictures, diagrams, gestures, computer graphics, written marks, and verbal expressions. However, these tools do not simply convey meanings; they become a channel or medium through which learners can construct meanings. These understandings are actively interpreted through semiotics (Chandler, 1994).
A semiotic approach to fraction instruction allows the teacher to use gestures and manipulatives, lead classroom discourse, and choose symbols that encourage students to construct an understanding of fraction size and equivalence. Semiotic theoretical perspectives have received much attention in recent years and have been applied widely in various mathematics instructional settings (Presmeg, 2006). The fundamental idea is to arrive at meaning through active interactions and activities that link together several semiotic representations, helping learners organize their actions and thoughts across space and time (Radford, 2003). Thus, the educational implications are to focus on multiple ways of accessing the curriculum, especially through discourse and using tools that will increasingly enable students to understand the meaning behind symbols (Cobb, 2000). By carefully designing a series of activities to engage students in visual, speaking and gesturing actions, we created a process whereby students progressively experience fraction size and equivalency. Radford, Bardini and Sabena (2007) explained that “to make something apparent, learners and teachers use signs and artifacts of different sorts (mathematical symbols, graphs, words, gestures, calculators, and so on). We call these artifacts and signs used to objectify knowledge semiotic means of objectification” (p. 5). Interestingly, especially due to the resources chosen here to engage students, Radford et al. referred to the resources and the synchronized activities that the students engage in as a “semiotic symphony” (p. 20). In this way, a teacher can guide students toward an initial conceptual understanding of fraction size and equivalence that can inform students’ use of more formal mathematical procedures.
2.2 Semiotics and multimodal instructional design
We designed our intervention on a semiotic framework, where the teaching-learning process includes both students’ and teacher’s use of semiotic resources in a multimodal way to stimulate a semiotic game (Arzarello, Paola, Robuti & Sabena, 2009). The notion of a semiotic game in the math classroom is grounded in Vygotsky’s conceptualization of the zone of proximal development where the teacher structures interactions to advance a student’s individual understanding and use of signs toward a more formal mathematical sense (Arzarello & Paola, 2007). In their notion of a semiotic game, Arzarello and Paola described the teacher’s role as a semiotic mediator who facilitates students’ internalization processes through signs. Semiotic resources used in teaching mathematics include oral and written words, extralinguistic modes of expression, inscriptions, and other tools or instruments used in the teaching process. Arzarello et al. (2009) described instruction in the classroom as a semiotic bundle, the range of semiotic activities that creates a system of signs, which develops as students solve problems and discuss mathematical questions with teacher assistance.
By utilizing a semiotic game approach to our instructional design, we created an environment filled with multimodal opportunities for students to make conceptual connections between properties inherent in fraction representations and fraction symbols. Because an effective learning environment should include opportunity and strategically selected tools for students to make connections and construct meaning (Abrahamson, 2009), we used music notation as the initial medium through which students grappled with their emerging understanding of fraction concepts. Through the use of language and gestures, students engaged in a semiotic game to create an understanding of fraction size and equivalence.
Academic Music is designed on a semiotic theoretical platform so that music notes and related clapping and drumming to create rhythms serve as the social medium through which students can explore the differences between whole numbers and fractions. Bringing music into the math classroom serves two purposes. First, this multimodal approach to early fraction instruction could encourage a deeper understanding of fraction reasoning because students are introduced to fraction concepts in fun and engaging ways. Our intervention relates mathematics to music by showing the relationship of musical rhythms to different sizes of fractions. Students learn to read musical notes and perform basic rhythmic patterns through clapping and drumming. They work toward adding musical notes together to produce a real number (fraction), and create addition/subtraction problems with musical notes. Activities are designed to reinforce note values by drumming rhythms based on the value of a whole note, and adding and subtracting fractions as these values relate to musical notes and rhythms. Second, schools are under such pressure to demonstrate adequate academic progress, administrators often reduce or eliminate music so more time can be devoted to traditional reading and mathematics instruction. In the USA, 20 % of school districts surveyed had greatly reduced instructional time for music (Center on Education Policy, 2005). In California, participation in general basic music classes between 1999 and 2004 declined by 85.8 %, representing a loss of 264,821 students (Music For All Foundation, 2004). If fraction instruction could be combined with genuine music instruction to teach math, educators could keep music in the classroom while addressing a central mathematical concept.
3 Research questions
This study examines the efficacy of a music intervention to teach fractions to third graders from a multicultural, mixed socio-economic public school setting. We are interested in three research questions. First, can this program teach introductory music notation so that students can use notes as a kind of manipulative? Second, can students transfer the fraction reasoning required for understanding music notation to fraction symbols and the related concepts of fraction size and equivalency? Third, will the effects of gaining conceptual understanding of fraction size and equivalency improve students’ performance in fraction computation?
The first question concerns how well students learn the basic music notation. With regard to this question, students will utilize the informal experience of adding and subtracting the temporal fraction values inherent in music notation to create music measures in the four fourths key signature. We hypothesize that this will mediate more formal fraction reasoning required for computation. The second question examines changes in conceptual knowledge between the experimental and comparison groups as determined by a test that measures understanding of fraction applications. Finally, the third question is answered in two ways. First, we will investigate any differences in the scores between the experimental group and the comparison group on a final fraction worksheet, developed by the research team. Second, we look at differences in starting abilities between the two groups, some of whom came to instruction with less music notation and fraction conceptual knowledge, to determine who benefitted from this intervention.
4 Methodology
4.1 Participants
The participants were 67 third-grade students (ages 8.5–10.11) from one kindergarten through eighth-grade elementary school in Northern California, with 94 % of the students identified as Hispanic or Latino and 68 % considered ELLs. Third graders from this school performed below the state average in reading and math on the California Achievement Test with only 28 % of reading scores and 48 % of math scores at or above the 50th percentile. The participants were all enrolled in four general education classrooms, including students with learning disabilities, attention deficit hyperactivity disorder, and hearing and speech impairments. The experimental group (n = 37) took part in the experimental mathematics instructional program, academic music. The comparison group (n = 30) continued regular mathematics instruction with their classroom teachers. Academic music was administered during regularly scheduled mathematics instruction so that the experimental students did not receive more mathematics instruction than their peers in the comparison classrooms.
Chi-square tests performed across experimental conditions revealed no significant differences among experimental and comparison groups in gender composition, ethnicity, age, language, or disability status. Independent samples t tests comparing the mean scores of the experimental and control group on the CELDT Overall Standard Scores, the CST Language Standard Scores, and the CST Math Standard Scores revealed no significant differences, t(60) = 1.09, p = .14, t(63) = .53, p = .83, t(62) = .27, p = .39, respectively.
4.2 Sampling procedures
Prior to the study, school administrators placed the students in one of four classrooms to create two intact classrooms of ELLs and two intact classrooms of students with proficient English Language skills. Random assignment of students to comparison and experimental groups was not logistically possible. Consequently, the school principal assigned one ELL class and one English-proficient class to the experimental condition.
4.3 Experimental design and data analysis
The study utilized a quasi-experimental comparison group pretest/posttest design (Cook & Campbell, 1979). First, we determined no significant differences in student demographics across experimental and comparison groups using Pearson’s chi-squared test. Second, independent samples t-tests compared mean scores of experimental and comparison groups on the CST and the CELDT to establish no significant differences between the mean scores of the two groups on academic achievement and English language proficiency prior to the start of the study. Third, to examine differences between comparison and experimental students’ posttest performance on the music test and the fraction concepts test, we planned to conduct analyses of covariance (ANCOVA) on the mean scores of each test using the pretest scores as the covariate to control the source of variation due to preintervention knowledge of music concepts and fraction concepts respectively. However, prior to performing the ANCOVAs, we tested the assumptions of equal slope using general linear regression. In both cases, this assumption was violated, so we examined the scatter plots and best fit lines to interpret the interaction. Then, to examine differences between the experimental group’s versus the comparison group’s performance on posttests, we performed independent samples t tests. We also performed an independent samples t test to examine differences between the groups on the mean scores of the final fraction worksheet. Finally, we performed an error analysis on the final fraction worksheet to examine differences in patterns of errors across groups.
5 Experimental procedures: instructional components of academic music
To design our instructional intervention, we employed components of the Kodaly system of music education because it necessitated experiential learning via several learning modalities (i.e., visual, auditory, and kinesthetic; Gault, 2005; Hurwitz, Wolff, Bortnick & Kokas, 1975).^{1} The language of the Kodaly method fitted into the semiotic game and enabled us to reinforce the proportional values of the notes with words. We provided opportunities for students to speak, move and gesture, allowing them to build an understanding of the notes and fraction symbols attached to them.
5.1 Academic Music intervention
Academic Music instruction
Lesson | Student objectives |
---|---|
1 | To recognize and understand whole, half, and quarter notes, their temporal size (number and length of beats), and their names (i.e., ta, ta-a, etc…) |
To understand the equivalent relationships between notes | |
To read music patterns (ostinatos) as measures in 4/4-time without error | |
2 | To understand equal beats as equal parts of the whole note |
To explain the length of whole, half, quarter and eighth notes | |
To write fundamental rhythm notation in 4/4 time | |
3 | To recognize and understand quarter and half note rest |
To add the values of notes and rests | |
4 | To write music patterns in 4/4 time by adding the values of notes and rests |
To demonstrate understanding of equal beats as equal parts of the whole note by correctly drumming music patterns in 4/4 time | |
5 | To write music patterns in 4/4 time by adding and subtracting the values of notes and rests |
6 | Review |
7 | To understand the meaning of the equal sign and equivalence |
To connect fraction names and symbols to music notes | |
To connect the magnitude of fractional quantities to the proportional value of music notes | |
To connect the equivalence of the proportional quality of music notes and rests to fractional quantities using fraction symbols | |
8 | To connect fraction bars and circles to proportional quantities in music notes |
To demonstrate understanding of equal parts of the whole with fraction bars and circles | |
To connect the addition of fractions with unlike denominators to the addition of different music notes | |
9 | To connect proportions of the number line to proportional quantities in music notes |
To utilize the number line to demonstrate proportioning the whole into fractional quantities | |
To utilize the number line to add and subtract fractions with unlike denominators | |
10 | To utilize fraction bars, circles, and the number line to add and subtract fractions |
11 | To add, subtract and multiply fractions with and without unlike denominators by choosing their own form of representation if needed |
12 | To demonstrate the understanding of fraction equivalence by correctly adding and subtracting fractions with unlike denominators |
During the first six lessons, the students received authentic music instruction, creating natural musical rhythms instead of just filling measures to add up to the value of the whole note. Rhythm relates to the relative time durations of notes and in most music a note is not sounded for more than a second (David, 1995). Rhythms in music are like the natural rhythms in our bodies like breathing and heartbeats and relate naturally to motion. David Jr. (1995) describes rhythm as movement in time and motion as movement in space. The standard of measurement in musical time is the beat and beats are generally grouped into sets of 2, 3, or 4 called measures or bars. We designated four beats in each measure with a time signature of four fourths time. The first two lessons followed the same sequence. The music teacher began by demonstrating how to step, clap and/or drum four lines of music, each containing four measures of note patterns. First, students learned the whole note by holding a clap and taking four steps of equal value. The gesture of holding the clap, counting 4 s, and taking four steps conveys quantifying information about the designated whole from which we could create other rhythms. Holding the clap for 4 s signifies the amount of time that creates a whole note and taking four steps presents additional information about the length or size of the whole. Then, as new notes were introduced, students first clapped and stepped, then drummed the notes’ time values. Through each activity, students are experiencing the varied amounts of time and length inherent in each note in our time signature as they relate to a fraction of the whole. In addition, through rhythm, students experience the note’s fractional value of the whole as continuous quantities that would later help the move to a number line representation. As instruction progressed, the music teacher continued to demonstrate how to drum the patterns by using drumsticks and saying the Kodaly syllabic name (see Wheeler, 1985) and the fractional name of each note. Students then used their drumsticks to drum the patterns as they simultaneously repeated the names of the notes. In the next four lessons, these music demonstrations became a review followed by instruction of the day’s fraction concepts. During this time, students were learning to add and subtract the proportional values of music notes to complete measures of music. They practiced arranging varied repeated patterns of notes (ostinatos) in each measure of four fourths time (e.g., one whole note, two half notes, one half note and two quarter notes). We used terms that conveyed the proportional quality of the notes such as “the whole note equals four of four equal beats” and “the quarter note equals one of four equal beats.” This language facilitated the introduction of fraction language and symbols during the following six lessons. Throughout instruction, we used this transparent fraction language that described the quantity along with the name of each fraction to increase conceptual understanding (Miura, Okamoto, Vlahovic-Stetic, Kim & Han, 1999; Paik & Mix, 2003). We utilized a semiotic framework and the concrete-pictorial-abstract approach (CPA; Bruner, 1977) to design the elements that comprised our instruction. Each lesson also included the sound instructional practices of teacher modeling, guided practice, independent practice and cumulative review (Scarlato & Burr, 2002).
The music teacher and the university researcher taught the next six lessons together. In this way, the university researcher was able to guide students toward connecting music instruction to more formal mathematics language, representations, and symbols. These lessons focused on explaining and demonstrating the concepts of fraction size and equivalence utilizing the connections students had made with music notes and their corresponding proportional values. Continuing the CPA sequence (Bruner, 1977), we presented repeated note patterns in measures of four fourths time, paying careful attention to maintain the melodic quality of the patterns. First, students were taught the fraction name and symbol that corresponded with each note. As students began to connect the fraction name to the note and fraction symbol, the instructors assisted students with making connections between representations and moving them toward more abstract, formal fraction symbols. At the end of each lesson, students completed a daily worksheet including such tasks as writing the fraction symbol that corresponds to each note in several measures of music and adding their values to be sure each measure contained four beats (see Fig. 2).
Starting with the seventh session, fraction circles, fraction bars, and the number line were introduced in succession so that students had an opportunity to partition varied representations of wholes into equal parts. The compare-and-contrast structure was used to help students identify the similarities and differences between partitioning different representations into fractional quantities (i.e., music notes versus the number line and fraction tiles; Miller & Hudson, 2007). Students compared the music notes’ values and fraction symbols to the corresponding portions of each representation. For example, eight eighth notes could fit between zero and one on the number line and in each measure of music. After comparisons were demonstrated, students completed their own representations of fraction bars and number lines using notes and fraction symbols. Finally, using fraction bars, the number line and notes, we taught students to “trade in” notes or “break up” the larger notes or fraction values into the smallest quantity that was in the computation problem. In this way, we taught addition and subtraction of fractions with unlike denominators. By focusing on the symbolic representation of fractions in musical rhythms, we never referenced or taught the procedural algorithm (e.g., cross-multiplication) for converting unlike denominators. Students completed fraction symbol problems with the option to use a chosen representational figure (i.e., music notes, fraction bars or number line). In both experimental classrooms, the regular classroom teachers remained in the classroom for the duration of the academic music sessions to observe and manage behavior problems if necessary. There were no behavior problems. However, on three occasions, the teacher in the Spanish-speaking classroom clarified directions for the class by translating them to Spanish. Outside of these occasions, we administered all sessions in English. Throughout instruction, we used transparent fraction language that described the quantity along with the name of each fraction to increase conceptual understanding (Miura et al., 1999; Paik & Mix, 2003). Embedded throughout the lessons were phrases such as “half, one of two equal parts” and “quarter, one of four equal parts.”
Utilizing components of the Kodaly method of music instruction, students were taught the temporal value of the whole, half, quarter, eighth, and sixteenth notes. The whole note was introduced as our invariant whole, which was the length of four beats. The other notes were introduced as portions of the whole. First, students learned the whole note by holding a clap for 4 steps of equal value. Then, as new notes were introduced, students first clapped and stepped, then drummed the notes’ time values. Students learned to arrange varied repeated patterns of notes (ostinatos) in each measure of four fourths time (e.g., one whole note, two half notes, one half note and two quarter notes). We used terms that conveyed the proportional quality of the notes such as “the whole note equals four of four equal beats” and “the quarter note equals one of four equal beats.” This language facilitated the introduction of fraction language and symbols during the following six lessons.
Each lesson followed the same instructional sequence. First, we handed out drumsticks and drum pads (inexpensive mouse computer pads) and “played” the music measures by drumming. As students tapped with the drumsticks, they said the Kodaly name for the notes. This activity served as a review of the notes and values and engaged the students in the lesson. Second, we reviewed the prior lesson and introduced the day’s objectives. For example, after the students were introduced to whole, half, and quarter notes, we introduced the symbol and meaning of a rest. Third, as we introduced and demonstrated new material, we provided opportunity for guided practice as a whole class. Working in pairs, students would complete a worksheet of problems that required them to connect values to notes, complete measures, connect notes to various pictorial representations of similar values. Fourth, we carefully monitored students and provided feedback as they practiced daily tasks in small groups and then independently. Finally, each day, classroom worksheets were collected and examined for student understanding and accuracy. These formative assessments enabled us to prepare effective review for the next session.
5.2 Instruction in both conditions
Both conditions (comparison and experimental) received the district mathematics curriculum. Students in the comparison group were taught by their regular classroom teachers for approximately 60 min/day, 5 days/week, using the state adopted curriculum and district texts. The text used in the district covered the California third grade math standards and focused on comparing fractions represented by drawings or concrete materials to show equivalency. In addition, the text explains addition and subtraction of simple fractions in context (e.g., one half of a pizza is the same amount as two fourths of another pizza that is the same size; show that three eights is larger than one fourth) (California Department of Education, 1999). Teachers in the control classrooms used the math texts and related workbook sheets to teach and practice adding and subtracting simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2) (California Department of Education, 1999). Students in the experimental group were taught by their regular classroom teachers for approximately 60 min/day, 3 days/week using the state adopted curriculum and district text. The other 2 days/week, students in the experimental class received the academic music intervention from the researchers during regularly scheduled math class. Both conditions received the same amount of mathematics instructional time over 6 weeks.
5.3 Classroom teachers
The regular classroom teachers in the control classroom designed and implemented fraction instruction using the district textbook. A supplemental workbook that accompanied the textbook provided teachers with classroom practice and homework worksheets for students. Classroom instruction and worksheets focused on equivalent fractions by demonstrating representations of parts of a whole and parts of a set. In addition, these representations were also used to demonstrate addition and subtraction of simple fraction. The regular classroom teachers in the academic music classes remained in the classroom as required by the school principal but had no role in lesson design or implementation.
5.4 Treatment fidelity
We audiotaped the implementation of each lesson across the two experimental classrooms. A fidelity checklist of crucial instructional points was constructed for each lesson using the daily lesson plan. Crucial instructional points from each part of the lesson were identified and described on a fidelity checklist. These instructional points corresponded with the required teacher behaviors for each part of the lesson (e.g., review of past lesson, introduction of daily objectives, music demonstration, and presentation of new material). Fidelity of treatment was calculated by equitably sampling 20 % of the tapes from each classroom. Fidelity scores were calculated based on the number of crucial instructional points covered in the script using the following formula: number of main points covered divided by number of total main points × 100 = percentage of main points covered. Fidelity of treatment in the Spanish-speaking classroom was 96 % (SD = 0.54), and in the English-speaking classroom was 98 % (SD = 0.44).
5.5 Measures
5.5.1 Music test
The music test included items that required students to identify music notes, match the fraction that corresponds to the value of the note, and add and subtract notes and fractions to maintain four fourths time in each measure (see Fig. 4). Maximum score for this measure was 33. Cronbach’s alpha was .89. Interscorer agreement, calculated by dividing the number of agreements by the number of agreements plus disagreements multiplied by 100, was computed on 20 % of pre- and posttests by two independent blind scorers. Interscorer agreement was 96 and 94 % for pre- and posttest, respectively.
5.5.2 Fraction concepts test
The fraction concepts test included items using regional areas, line segments, and set representations to check understanding of the part-whole concept of fractions and related equality as well as fractions as numbers on the number lines (Ni, 2001; Niemi, 1996; see Fig. 5). To demonstrate understanding of fraction concepts, the test required students to move between the fraction symbol and the fraction representation (Behr, Lesh, Post & Silver, 1983; Hiebert, 1989; Ni, 2001). Maximum score was 21 with six questions providing opportunity for more than one answer (e.g., see Fig. 5, question 3). Cronbach’s alpha was .82. Interscorer agreement was 90 % for both pre- and posttest.
5.5.3 Fraction worksheets
The first fraction worksheets contained fraction computation problems with like denominators accompanied by representations. The difficulty level of the worksheets gradually increased to fraction-symbol-only problems with like and unlike denominators (see Fig. 6). The final worksheet included fractions not previously introduced (e.g., one third) and improper fractions, and was administered to both control and experimental groups. While both groups were accustomed to completing worksheets in class, they had not completed identical worksheets prior to completing the final worksheet. The purpose of this final worksheet was to examine whether students’ conceptual understanding of fraction size and equivalency transferred to an ability to compute familiar and unfamiliar fractions without representations for reference. There were ten problems on the worksheet with four problems requiring students to find a common denominator. Regarding those four problems, students received a point for identifying the correct denominator and a point for correct computation. The maximum score was 14. Interscorer agreement, computed on 20 % of the final worksheets as described above, was 95 %.
5.6 Data collection
The authors administered the music test, the fraction concepts test, and the last fraction computation worksheet to all students in class, as a group. Pretesting on the music test and the fraction concepts test took place 1 week before the start of the first lesson. The posttest of the music test and the fraction concepts test took place 1 week after the last lesson was completed. The last fraction computation worksheet was administered directly after instruction and after the posttest for the comparison group. At pre- and posttest, we read aloud the directions and sample problems for each section and provided students with time to complete work before progressing to the next section. We continued to the next section when all but one student appeared to be finished.
6 Results
6.1 Analysis of treatment effects
Performance data
Variables | Comparison | Experimental | Comparison | Experimental |
---|---|---|---|---|
n = 30 | n = 37 | Subgroups^{a} | Subgroups^{a} | |
X (SD) | X (SD) | X (SD) | X (SD) | |
Music test | n = 16 | n = 16 | ||
Grand X (SD) for pretest 3.94 (2.56) | ||||
Pretest | 3.77 (2.64) | 4.08(2.53) | 1.69 (1.01) | 1.75 (1.13) |
Posttest | 6.23 (4.13) | 15.78(8.46)* | 5.44 (3.85) | 11.13 (7.01)* |
Fraction concep ts test | n = 10 | n = 15 | ||
Grand X (SD) for pretest 8.03 (3.36) | ||||
Pretest | 7.87 (2.66) | 8.16 (3.86) | 5.00 (1.76) | 4.47 (2.39) |
Posttest | 12.20 (3.09) | 13.43 (2.51) | 9.10 (2.28) | 11.60 (2.26)* |
Fraction computation worksheet | ||||
Posttest | 3. 87 (2.21) | 7.0 8 (3.88)* | 3. 30 (2.79) | 5.67 (4.79)* |
6.1.1 Music notation knowledge
6.1.2 Fraction concepts test
6.1.3 Fraction computation
Percentage of correct responses for computation problems by group
Entire sample | Smaller sample^{a} | ||||
---|---|---|---|---|---|
Experimental | Comparison | Experimental | Comparison | ||
# | Problem | n = 37 | n = 30 | n = 15 | n = 10 |
1 | 1/2 + 1/2 | 89 | 77 | 73 | 50 |
2 | 1/2 − 1/4 | 43 | 7 | 33 | 10 |
3 | 2/4 + 4/8 | 43 | 3 | 47 | 10 |
4 | 4 × 1/4 | 70 | 53 | 53 | 50 |
5 | 1/3 + 1/3 | 81 | 73 | 53 | 50 |
6 | 4/4 − 1/4 | 68 | 53 | 53 | 50 |
7 | 1/4 + 1/4 + 1/8 | 8 | 0 | 7 | 0 |
8 | 2 × 1/2 | 70 | 60 | 53 | 50 |
9 | 6/4 − 1/2 | 19 | 0 | 13 | 0 |
10 | 3 × 1/3 | 76 | 50 | 60 | 40 |
Error analysis
Error type | Experimental group (n = 37) | % of errors | Comparison group (n = 30) | % of errors |
---|---|---|---|---|
Correct denominator, incorrect conversion | 36 | 20 | 49 | 23 |
Incorrect denominator, incorrect conversion | 38 | 21 | 11 | 5 |
Correct denominator, incorrect computation or disregard to sign | 12 | 7 | 5 | 2 |
No answer, required conversion or multiplication | 20 | 11 | 41 | 19 |
No conversion required for multiplication, conversion incorrect | 1 | 0.5 | ||
Multiplied numerator and denominator with whole number and fraction (2 × 1/2) or multiplied denominator only/incorrect multiplication | 13 | 7 | 25 | 12 |
No attempt, drew one note for each problem or just no attempt | 12 | 7 | 16 | 8 |
Drew pictures to help solve problem, but incorrect | 7 | 4 | 4 | 2 |
Add/subtract numerator and/or denominator | 15 | 8 | 52 | 24 |
Incorrect, no discernable pattern | 28 | 15 | 10 | 5 |
Total mistakes | 182 | 213 | ||
Total possible | 518 | 420 | ||
Group accuracy | 65 % | 51 % |
7 Discussion
By introducing a number of fraction representations in a stepwise fashion across the academic music intervention, we created a semiotic chain (Presmeg, 2006), a sequence of abstractions that started with music notation (the first signs) and then introduced discrete and continuous representations of fractions (the second signs). We completed the chain with formal fraction symbols (the third signs). In this way, we developed students’ understanding of fraction size and equivalence in a series of steps that started with musical activities and progressed toward more formal mathematical concepts of fraction equivalence and size. We will begin by addressing the research questions and the subsequent theoretical underpinnings that explain our results. Second, we will discuss the social validity of the academic music intervention, study limitations, and future research possibilities. Finally, we will summarize our findings and present concluding remarks.
7.1 Main findings
7.1.1 Music notation
The outperformance of the experimental group over the comparison group on the music test suggests that the experimental group acquired significantly more music notation knowledge than the comparison group, who did not receive this type of instruction. In fact, the mean score of students in the experimental group was almost one and a half standard deviations over the mean score of students in the comparison group at posttest. This disparity most likely reflects the design of the study in which the comparison group only receives standard fraction instruction while the experimental group had an additional music component.
Academic music required students to demonstrate their understanding of the value of notes with clapping, drumming, reciting names, and adding and subtracting the values of notes to complete measures. The music notes acted as the first signs or representation in the semiotic chain leading to more formal mathematic symbols. The instructional gestures served as signs or representations for fractional temporal units and provided repeated opportunities for students to construct and internalize the size of the related fractional units. Then, as instruction moved toward more formally connecting the musical concepts to fractions symbols, students may have already internalized their understanding of size due to the experience of drumming the temporal values of the notes. It may have been the daily acts of drumming and saying each note’s name in Kodaly syllables that provided a kinesthetic, oral, and visual mode for experiencing temporal values of notes and related fractions. The gesture of students holding a clap or holding the drumstick still for four counts and saying ta-ah-ah-ah as a representation of the whole note provided opportunity for them to experience the value of the whole. Notes of shorter duration, like the half note and quarter note, provided opportunity for students to experience more claps or drums with notes of shorter duration. We suggest that this experience helped students to more readily understand that the smaller the denominator (note value), the bigger the numerator (number of notes) to equal the value of the whole. In turn, holding the clap or drumstick for four counts and breaking the measure up with several smaller notes provided a natural transition to using the number line when representing fractions.
7.1.2 Transfer of fraction understanding
Moving between representations of fractional quantities and fraction symbols is one of the most difficult aspects of fraction instruction (Hiebert, 1989; Mack, 1995). However, utilizing physical gestures and various representations associated with fractional quantities appeared to help struggling students as they worked to identify, add, and subtract notes and fractional quantities to create measures in four fourths time (see Figs. 1, 2, and 3). Bruner emphasized the use of three distinct kinds of materials in teaching mathematics to children that tie in to the CPA sequence: enactive, iconic, and symbolic (Ediger, 1999). Enactive materials accentuate the use of concrete models for children to learn using a hands-on approach through manipulation of the materials. Iconic materials correspond to pictoral representations and highlight the use of pictures, illustrations, and other visual aids. Symbolic materials are more abstract and emphasize the use of textbooks and more formal mathematical content. The underlying idea is to begin instruction using concrete models and move progressively through more abstract ones, helping students build strong connections along the way.
Our concrete level of instruction involved clapping, stepping and drumming the value of music notes, using hands and drumsticks as manipulatives. The pictorial level involved students appropriately filling music measures in four fourths time with varied patterns of notes or conversely removing notes to maintain the four fourths time within each measure. In addition, the value of notes was alternately compared to parts of fraction circles, fraction bars and places on the number line. The abstract level of instruction involved moving students away from musical notes and fractional representations to using only fraction symbols to add, subtract, and multiply fraction quantities.
The music test (see Fig. 4) and fraction concept test (see Fig. 5) measured transfer of fraction reasoning between music notation and fraction symbols. Overall, students in the experimental group significantly outperformed students in the comparison group on the music test. Students in the experimental group demonstrated the ability to transfer knowledge of music notes to fraction symbols. They were also able to add and subtract proportional values of the notes and fraction symbols to create measures that fit into the key signature. The ability to create measures in four fourths time with both notes and fraction symbols showed that these students were gaining an emerging understanding of fraction reasoning. As measured by the fraction concepts test, both students in the experimental and comparison groups made significant improvements in their conceptual understanding of basic fractions. Because all third grade students were being exposed to introductory fraction concepts in the standard curriculum, the mean scores of all students improved from pre- to posttest. While we found no significant mean difference between the students in the experimental group and the comparison group on the fraction concepts posttest, examination of the scatterplots suggests two explanations. First, this test aligns more closely with the standard curriculum, so all students taught through that curriculum would be expected to make similar progress. Second, students in the experimental group who came to academic music with less conceptual knowledge than their lower achieving peers in the comparison group (based on pretest fraction concept scores) may have utilized their emerging fraction reasoning knowledge to outperform their peers in the comparison group at posttest. In addition, students who performed below the grand mean in the experimental group scored over one standard deviation higher than their peers in the comparison group. While the sample is small and we must be careful not to overgeneralize, the students who came to instruction with less conceptual knowledge at pretest appeared to gain significantly from this instruction. Lower performing students often require more intensive interventions than whole class instruction to show evidence of emerging understanding, so we are encouraged by their positive response to academic music.
7.1.3 Fraction computation
Students in the experimental group outperformed students in the comparison group on the final fraction worksheet with a strong effect size. The mean score for students in the experimental group was about one standard deviation higher than the mean score of students in the comparison group. Students in the experimental group clearly outperformed their peers in the comparison group on every item, but there are important differences in the types of errors made. For example, students in the experimental group were far less likely to add across numerator and denominator, a common mistake made by students learning to add and subtract fractions, which often persists into high school (Newton, 2008). Twenty-four percent of total errors made by the comparison group compared to 8 % of total errors made by the experimental group were due to adding or subtracting across numerator and denominator. Mack (1990) suggests that teaching formal algorithms and rote procedures before students have developed conceptual understanding of basic fraction concepts may cause students to blindly apply procedures. Because only 8 % of the total errors made by students in the experimental group were due to adding or subtracting the numerator and the denominator, those students may have utilized the connections they made between the fractional value of notes and equivalent fraction symbols to inform their computation. When adding or subtracting fractions with unlike denominators, students in the experimental group averaged about 28 % correct while students in the comparison group averaged about 3 % correct.
Students in the experimental group may have utilized their conceptual understanding of the proportional sizes of music notes and related fraction symbols to inform computation of unfamiliar fractions. By contrast, students in the comparison group may have relied on procedural knowledge to calculate without fully understanding proportions. These explanations are based on an error analysis of the final fraction worksheet (Table 4). This worksheet included items not taught or included on prior worksheets so that we could look for evidence of transfer (see bold items in Table 3). Furthermore, the percentage of total errors due to leaving an item that required conversion or multiplication blank was 19 % by students in the comparison group versus 11 % by students in the experimental group. Students in the experimental group seemed to take more risks and attempt difficult fraction problems more readily than their peers in the comparison group. As students become more willing to make an effort to answer mathematics questions, they may gain confidence, lessen their mathematics anxiety, and improve their performance (Ashcraft & Krause, 2007; Furner & Berman, 2004; Shields, 2005).
All students in this study were taught a procedural algorithm for computing fractions with unlike denominators in their regular math class. However, the comparison group may have relied only on the algorithm but they did not use it consistently. Though the comparison group’s performance on the fraction concepts test was comparable to the experimental group’s performance, it may have been that the comparison group was overly reliant on the learned procedure for computing with unlike denominators and less likely to utilize their conceptual knowledge to inform computation. While the experimental group demonstrated an understanding of the need to convert unlike denominators before adding or subtracting fractions, 21 % of their total errors were due to incorrect conversions (e.g., trying to convert smaller denominators to larger denominators for addition or subtraction). In examining the actual test protocols, diagrams and calculations suggest that students in the experimental group knew a conversion was required to compute unlike denominators but those individuals with more fragile understanding either drew notes larger than should be used or they made addition or subtraction errors. By contrast, fewer errors in the comparison group were due to incorrect conversions (5 %) because no attempt at conversion was made. Students in the comparison group appeared less aware of the need to convert unlike denominators. Students in the experimental group had more opportunities to practice proportional reasoning with unlike denominators during academic music instruction. So, even though some fractions had unfamiliar denominators (e.g., one third), they were more likely to attempt to transfer fraction reasoning to novel problems. The relative inability of the comparison group to transfer knowledge to novel problems suggests a more fragile conceptual understanding of fraction size and equivalency, and demonstrates the importance of providing more ample opportunities for students to solve novel and difficult fraction problems. However, the relatively low performance of all students on the transfer questions shows that the development of fraction skills can be slow and complex. With practice, students can develop the experience necessary to correctly apply conceptual knowledge and procedural strategies to more complex fraction problems (Brigham et al., 1996).
It appears that without a representational picture to show the size of the units involved in the problem, some students may have allowed their whole number knowledge to interfere with choosing the correct, or smallest, denominator to divide into equivalent parts for calculation. Mack (1990) found that when students are first introduced to fractions, they frequently confound whole number principles with newly developing fraction concepts, especially when they are using standard fraction symbols only, as in the case of the final fraction worksheet. While the comparison group was also taught to convert unlike denominators using a procedural algorithm without a conceptual component, they did not refer to a variety of representations (i.e., music notes, fraction circles, fraction bars, and the number line) as referred to by the experimental group. Also, the daily tasks of drumming to make the music sounds and the cognitive movement between a variety of representations in the experimental group may have provided adequate opportunities for students to begin constructing meaning for the size and equivalency concepts required to inform fraction computation (Arzarello & Robuti, 2004; Schnepp & Chazan, 2004; Rasmussen, Nemirovsky, Olszewski, Dost & Johnson, 2004).
7.1.4 Student differences
Overall, students in the experimental group made significant improvements from pre- to posttest on the music test and the fraction concepts test. Students in the comparison group also made significant gains from pre- to posttest on the fraction concepts test as the test was aligned with the curriculum in place at the school. However, experimental students who came to instruction with less conceptual knowledge than their higher achieving peers made significant gains over similar students in the comparison group on the fraction concepts test (see Fig. 8). This is an important finding because students in the experimental group who came to the intervention with less conceptual knowledge appeared to more readily construct an understanding of fraction concepts and surpassed similar students in the comparison group. Students in the experimental group appeared to benefit from the addition of academic music instruction to nearly meet the mean performance of their peers and also made significant gains over students in the comparison group at posttest. In the experimental group, students with pretest scores below the grand mean scored posttest scores over one standard deviation higher than the mean score for similar peers in the comparison group. While students in the control group benefitted from the regular curriculum to make significant gains on the Conceptual Knowledge test, those lower performing students in the group did not gain as much as the lower performing students exposed to academic music. One explanation may be that lower performing students benefit from the multimodal semiotic approach to instruction because of its increased physical and cognitive engagement.
7.1.5 Social validity
In order to evaluate the social validity of academic music, we examined our goals of treatment, procedures, and the outcomes produced by these procedures (Foster & Mash, 1999). The goal of academic music was to teach fraction concepts through authentic music instruction. In this way, we addressed an important instructional hurdle in the elementary curriculum, fraction concepts, and simultaneously provided music instruction, which is an overlooked core content subject in the elementary curriculum. Our procedures included evidence-based instructional practices that are effective for all students, especially students with learning differences (Bruner, 1977; Ediger, 1999; Miller & Hudson, 2007).
Our procedures were clearly outlined in lesson plans and could be put together with instructions to create an easy-to-use manual. Both experimental teachers said they would and could use academic music in their classrooms with some training. The outcomes produced by our academic music procedures suggest these students gained a better conceptual understanding of fraction size and equivalency than students who spent all of their math instructional time learning from the standard curriculum. In addition, academic music brought music back into the classroom with genuine music instruction, which is lacking in many underfunded schools. We suggest that using music instruction to indirectly teach fraction concepts may serve as an important motivator for students to learn and persist in understanding difficult fraction concepts. In addition, teachers’ willingness to teach academic music suggests the program may address teachers’ discomfort when teaching difficult mathematics concepts (Frykholm, 2004) and the lack of conceptual instructional guidance in classroom materials and texts (Sood & Jitendra, 2007).
7.1.6 Study limitations
There were some limitations to our study. First, we had only two experimental and two comparison classrooms from the same school for a total of 67 participants. Second, students were not randomly assigned to comparison and experimental group, but assigned by classroom to conditions by the school principal. Third, approximately half of the students in this study were ELLs, so these results can only be generalized to similar populations of students. Because of the small sample size, lack of randomization, and specific sample characteristics, results may not occur with a different sample. Finally, while both teachers in the academic music group thought they could implement academic music in their classrooms, we cannot be certain that they would obtain similar results if they did not have some music training. Though we include specific instructions in lesson plans and worksheets, and the basis of the music component is rhythm, it is nonetheless important that the sound of the measures in four fourths time is music and not just four counts of sound.
8 Conclusions
While there are limitations to this study, the results show promise for the use of music to teach fraction concepts in the elementary curriculum. We have a compelling reason to view music instruction as an integral part of the elementary curriculum, due to its utility in teaching beginning fraction concepts and related fraction computation to elementary students. Furthermore, this intervention appears to be particularly effective for students who are coming to instruction with a lower than average understanding of fractions. Academic music appears to have strengthened their conceptual understanding of both the magnitude and equivalency of fractions via a semiotic game. Curriculum developers must keep in mind ways to address learners who do not initially respond to sound classroom instruction, with the goal to reach all learners.
Radford, Edwards and Arzarello (2009) describe in a special issue of Educational Studies in Mathematics how semiotic instruction can be effective for students at varied levels of understanding because it helps students make connections between multiple representations of important ideas and encourages interactions with ideas and expressions in new ways. In a more recent special issue on semiotics, Radford, Schubring and Seeger (2011) further describe how teaching and making meaning of difficult mathematics concepts have been influenced by this blossoming semiotic perspective. The authors suggest that meaning making can be a communal classroom activity system in which teachers “create zones of proximal development where students make meaning” (Radford et al., 2011, p. 155). In this way, semiotic instruction, using multimodal resources and gestures, may help increase conceptual understanding of fractions and proportions. Academic music may be used as both a way to introduce basic fraction concepts to an entire class, and as a tutoring intervention for reviewing fraction concepts in a small group.
Academic music was designed to introduce beginning fraction concepts to third-grade students with and without learning differences. As we focused on fraction symbols, fraction size and fraction equivalency, we taught students that when ordering fractions or finding equivalent fractions, the whole must be the same size. While teaching the invariance of the whole may have enabled students to more readily learn these beginning concepts, it may impede instruction later when students are confronted with a variable unit in problem solving. Because we utilized several representations of fractions, future instruction that introduces the variable unit can continue with several of these already familiar representations. For example, the number line is particularly well suited to introduce a variable unit because a length can represent a unit and the number line model allows for iteration of the unit and immediate subdivisions of all iterated units (Bright, Behr, Post & Wachsmuth, 1988). While we do not consider our use of an invariable whole to be a limitation, we do feel the need to point out that future instruction must address the notion that a fraction represents a relative amount.
Future research will include creating an academic music manual, with a CD or DVD with music examples, complete with lesson plans and instructions for teachers to use in their own classrooms. We hope that teachers can administer academic music to their students with integrity and fidelity to achieve similar results. Future studies will examine the effects of a more general program, rooted in the concept of semiotics, to increase global mathematics achievement of students in elementary school.
The Kodaly method originated in Hungary and stresses that music instruction should begin at an early age when children naturally love music and are freely able to absorb the inherent rhythm and timing. The Kodaly method incorporates a rhythm syllables system in which note values are assigned specific syllables, which literally express their durations. For example, quarter notes are expressed by the syllable “ta” while eighth note pairs are expressed using the naturally shorter syllables “ti-ti.” Larger note values are expressed by extending ta to become ta-a for the half note and ta-a-a-a for the whole note (see Wheeler, 1985).