academic music

Academic music: music instruction to engage third-gradestudents in learning basic fraction concepts

Susan Joan Courey & Endre Balogh &

Jody Rebecca Siker & Jae Paik

Published online: 23 March 2012# Springer Science+Business Media B.V. 2012

Abstract This study examined the effects of an academic music intervention on conceptualunderstanding of music notation, fraction symbols, fraction size, and equivalency of thirdgraders from a multicultural, mixed socio-economic public school setting. Students (N067)were assigned by class to their general education mathematics program or to receiveacademic music instruction two times/week, 45 min/session, for 6 weeks. Academic musicstudents used their conceptual understanding of music and fraction concepts to inform theirsolutions to fraction computation problems. Linear regression and t tests revealed statisti-cally significant differences between experimental and comparison students’ music andfraction concepts, and fraction computation at posttest with large effect sizes. Studentswho came to instruction with less fraction knowledge responded well to instruction andproduced posttest scores similar to their higher achieving peers.

Keywords Fraction concepts . Elementary . Representation . Music notation . Semiotics

1 Introduction

Fractions are one of the most difficult mathematical concepts to master in the elementarycurriculum (Behr, Wachsmuth, Post & Lesh, 1984; Cramer, Post & delMas, 2002; Moss &Case, 1999). For many students, the struggle to understand fractions continues throughmiddle and high school, thereby delaying or preventing development of mathematicalreasoning and mastery of algebraic concepts (Brigham, Wilson, Jones & Moisio, 1996;Mazzocco & Devlin, 2008; National Research Council, 2001). For students with learningdifficulties, English Language Learners (ELLs), and students with low achievement, mas-tering 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).

Educ Stud Math (2012) 81:251–278DOI 10.1007/s10649-012-9395-9

S. J. Courey (*) : E. Balogh : J. R. Siker : J. PaikSan Francisco State University, San Francisco, CA, USAe-mail: [email protected]

Fraction instruction, like other content areas in the elementary curriculum, has undergonetwo large changes brought on by the school reform movement. First, in response to past poorperformance of US students on international mathematics assessments (McLaughlin, She-pard & O’Day, 1995), the National Council of Teachers of Mathematics (NCTM) set forthspecific standards for mathematical reform (Maccini & Gagnon, 2002; NCTM, 2000).Included in the NCTM standards are specific goals and recommendations for curricularchanges regarding fractions and fraction instruction. More recently, many states are adoptingthe 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 becomeeven more important parts of the third-grade math curriculum. Second, with the implemen-tation of No Child Left Behind of 2001, its impending reauthorization, and the reauthoriza-tion of the Individuals with Disabilities Education Act (IDEA) in 2004, effectivemathematics achievement for all students within the least restrictive environment must berealized and all students must be prepared to meet proficiency goals on statewideassessments.

NCTM and Common Core standards related to fractions for elementary through middleschool emphasize conceptual understanding. In turn, reform-based fraction instruction hasshifted from more behavioral, procedurally driven, rule-based approaches to a more cogni-tive conceptual approach with emphasis on problem solving and reasoning (Butler et al.,2003; Woodward, Baxter & Robinson, 1999). Fraction size includes knowing a fraction 1/bas the quantity formed by 1 part when a whole is partitioned into b parts and understanding afraction a/b as the quantity formed by a parts of size 1/b. Understanding fraction size alsoincludes identifying a fraction on a number line. Equivalency refers to different fractionsrepresenting the same rational number. NCTM (2000) standards suggest that young childrenbe exposed to simple fractions through real-life experiences and language. These youngstudents should understand and be able to visually represent commonly used fractions suchas one half, one fourth, and one eighth. In grades three to five, students should (a) develop anunderstanding of fractions as parts of wholes, as parts of a collection, as locations on thenumber line, and as divisions of whole numbers, (b) use models, benchmarks, and equiv-alent forms to judge the size of fractions, and (c) recognize and generate equivalent forms ofcommonly used fractions. The Common Core fraction standards for third grade are similarbut 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 interms of one unit to another unit. To understand this, students must realize that the magnitude ofthewholes for equivalent fractions should be the same (Post, Behr& Lesh, 1986). Two fractionscan 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 ofthe whole, helps students to understand the order of fractions as quantities rather than assymbols. A fragile understanding of fractional parts is the barrier that prevents students frommastering computation skills with fractional quantities (Young-Loveridge, Taylor, Hawera &Sharma, 2007). Yoshida and Shinmachi (1999) suggest expressly teaching the equal-wholeschema for students to more easily understand the concepts of equivalence and magnitude. Inthe present study, we have designed a fraction instructional intervention that utilizes theproportional representation of fractional quantities inherent in musical notes.

Techniques and strategies to teach difficult introductory fraction concepts to groups ofheterogeneous students in general education classrooms should be designed with efficacyand ease of implementation in mind. An instructional strategy must address students’difficulty in shifting from whole to rational number reasoning with a focus on developingunderstanding of fractions as numbers (Mack, 1995). The instructional design must also

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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 isespecially important because each lesson has to be designed so that a teacher without musicinstruction experience or training could deliver the intervention with fidelity. We werecareful to include clear directions and worked examples with each lesson plan and work-sheet. In order for our intervention to have social validity (Foster & Mash, 1999), we wantedto be sure that teachers could use Academic Music without the help of a music teacher. Inaddition, we sought to make each lesson fun and entertaining for both the teacher and thestudents.

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 characteristicsand goals. However, novice learners may not readily perceive the meanings behind symbolicrepresentation. Thus, from an educational stance, the process behind how young childrenconstruct symbolic meanings becomes an important question. Influenced largely by Vygot-sky’s work (see Vygotsky, 1978), mathematical development has been viewed as learningtools and signs that are elements of specific communicative systems, which vary by culture.According to semiotic perspectives, symbolizing involves manipulations with cultural toolssuch as physical objects, pictures, diagrams, gestures, computer graphics, written marks, andverbal expressions. However, these tools do not simply convey meanings; they become achannel or medium through which learners can construct meanings. These understandingsare actively interpreted through semiotics (Chandler, 1994).

A semiotic approach to fraction instruction allows the teacher to use gestures andmanipulatives, lead classroom discourse, and choose symbols that encourage students toconstruct an understanding of fraction size and equivalence. Semiotic theoretical perspec-tives have received much attention in recent years and have been applied widely in variousmathematics instructional settings (Presmeg, 2006). The fundamental idea is to arrive atmeaning through active interactions and activities that link together several semiotic repre-sentations, helping learners organize their actions and thoughts across space and time(Radford, 2003). Thus, the educational implications are to focus on multiple ways ofaccessing the curriculum, especially through discourse and using tools that will increasinglyenable students to understand the meaning behind symbols (Cobb, 2000). By carefullydesigning a series of activities to engage students in visual, speaking and gesturing actions,we created a process whereby students progressively experience fraction size and equiva-lency. 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 usedto objectify knowledge semiotic means of objectification” (p. 5). Interestingly, espe-cially due to the resources chosen here to engage students, Radford et al. referred tothe resources and the synchronized activities that the students engage in as a “semi-otic symphony” (p. 20). In this way, a teacher can guide students toward an initialconceptual understanding of fraction size and equivalence that can inform students’use of more formal mathematical procedures.

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2.2 Semiotics and multimodal instructional design

We designed our intervention on a semiotic framework, where the teaching-learning processincludes both students’ and teacher’s use of semiotic resources in a multimodal way to stimulatea semiotic game (Arzarello, Paola, Robuti & Sabena, 2009). The notion of a semiotic game inthe math classroom is grounded in Vygotsky’s conceptualization of the zone of proximaldevelopment where the teacher structures interactions to advance a student’s individual under-standing 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 semioticmediator who facilitates students’ internalization processes through signs. Semiotic resourcesused in teaching mathematics include oral and written words, extralinguistic modes of expres-sion, 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 semioticactivities that creates a system of signs, which develops as students solve problems and discussmathematical questions with teacher assistance.

By utilizing a semiotic game approach to our instructional design, we created anenvironment filled with multimodal opportunities for students to make conceptual connec-tions between properties inherent in fraction representations and fraction symbols. Becausean effective learning environment should include opportunity and strategically selected toolsfor students to make connections and construct meaning (Abrahamson, 2009), we usedmusic notation as the initial medium through which students grappled with their emergingunderstanding of fraction concepts. Through the use of language and gestures, studentsengaged 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 andrelated clapping and drumming to create rhythms serve as the social medium through whichstudents can explore the differences between whole numbers and fractions. Bringing music intothe math classroom serves two purposes. First, this multimodal approach to early fractioninstruction could encourage a deeper understanding of fraction reasoning because students areintroduced to fraction concepts in fun and engaging ways. Our intervention relates mathematicsto music by showing the relationship of musical rhythms to different sizes of fractions. Studentslearn to read musical notes and perform basic rhythmic patterns through clapping and drum-ming. They work toward adding musical notes together to produce a real number (fraction), andcreate addition/subtraction problems with musical notes. Activities are designed to reinforcenote values by drumming rhythms based on the value of a whole note, and adding andsubtracting fractions as these values relate to musical notes and rhythms. Second, schools areunder such pressure to demonstrate adequate academic progress, administrators often reduce oreliminate music somore 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 classesbetween 1999 and 2004 declined by 85.8%, representing a loss of 264,821 students (Music ForAll Foundation, 2004). If fraction instruction could be combined with genuine music instructionto teach math, educators could keep music in the classroom while addressing a centralmathematical concept.

3 Research questions

This study examines the efficacy of a music intervention to teach fractions to third gradersfrom 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 studentscan use notes as a kind of manipulative? Second, can students transfer the fraction reasoningrequired for understanding music notation to fraction symbols and the related concepts offraction size and equivalency? Third, will the effects of gaining conceptual understanding offraction size and equivalency improve students’ performance in fraction computation?

The first question concerns how well students learn the basic music notation. With regardto this question, students will utilize the informal experience of adding and subtracting thetemporal fraction values inherent in music notation to create music measures in the fourfourths key signature. We hypothesize that this will mediate more formal fraction reasoningrequired for computation. The second question examines changes in conceptual knowledgebetween the experimental and comparison groups as determined by a test that measuresunderstanding 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 andthe 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 whomcame to instruction with less music notation and fraction conceptual knowledge, to deter-mine who benefitted from this intervention.

4 Methodology

4.1 Participants

The participants were 67 third-grade students (ages 8.5–10.11) from one kindergartenthrough eighth-grade elementary school in Northern California, with 94% of the studentsidentified as Hispanic or Latino and 68% considered ELLs. Third graders from this schoolperformed below the state average in reading and math on the California Achievement Testwith 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 studentswith learning disabilities, attention deficit hyperactivity disorder, and hearing and speechimpairments. The experimental group (n037) took part in the experimental mathematicsinstructional program, academic music. The comparison group (n030) continued regularmathematics instruction with their classroom teachers. Academic music was administeredduring regularly scheduled mathematics instruction so that the experimental students did notreceive more mathematics instruction than their peers in the comparison classrooms.

Chi-square tests performed across experimental conditions revealed no significant differ-ences among experimental and comparison groups in gender composition, ethnicity, age,language, or disability status. Independent samples t tests comparing the mean scores of theexperimental and control group on the CELDT Overall Standard Scores, the CST LanguageStandard Scores, and the CST Math Standard Scores revealed no significant differences, t(60)01.09, p0 .14, t(63)0 .53, p0 .83, t(62)0 .27, p0 .39, respectively.

4.2 Sampling procedures

Prior to the study, school administrators placed the students in one of four classrooms tocreate two intact classrooms of ELLs and two intact classrooms of students with proficientEnglish Language skills. Random assignment of students to comparison and experimentalgroups was not logistically possible. Consequently, the school principal assigned one ELLclass and one English-proficient class to the experimental condition.

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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 demographicsacross experimental and comparison groups using Pearson’s chi-squared test. Second,independent samples t-tests compared mean scores of experimental and comparison groupson the CST and the CELDT to establish no significant differences between the mean scoresof the two groups on academic achievement and English language proficiency prior to thestart of the study. Third, to examine differences between comparison and experimentalstudents’ posttest performance on the music test and the fraction concepts test, we plannedto conduct analyses of covariance (ANCOVA) on the mean scores of each test using thepretest scores as the covariate to control the source of variation due to preinterventionknowledge of music concepts and fraction concepts respectively. However, prior toperforming the ANCOVAs, we tested the assumptions of equal slope using general linearregression. In both cases, this assumption was violated, so we examined the scatter plots andbest fit lines to interpret the interaction. Then, to examine differences between the experi-mental group’s versus the comparison group’s performance on posttests, we performedindependent samples t tests. We also performed an independent samples t test to examinedifferences 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 inpatterns of errors across groups.

5 Experimental procedures: instructional components of academic music

To design our instructional intervention, we employed components of the Kodaly system ofmusic 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 toreinforce the proportional values of the notes with words. We provided opportunities forstudents to speak, move and gesture, allowing them to build an understanding of the notesand fraction symbols attached to them.

5.1 Academic Music intervention

Academic music instruction included twelve 40-min sessions, delivered twice per week for6 weeks by a music teacher and a university researcher (see Table 1). The first six lessonsfocused on music notation and the temporal value of music notes in four fourths time. Bytemporal value, we refer to the relative time duration of each note. The last six lessonsfocused on connecting the proportional values of the music notes to other signs or fractionrepresentations and then to formal mathematical fraction symbols. The sequence of instruc-tion was as follows: first, students were taught basic music notation for measures in fourfourths time and how many beats of each note was equal to a whole note, the largest quantity

1 The Kodaly method originated in Hungary and stresses that music instruction should begin at an early agewhen children naturally love music and are freely able to absorb the inherent rhythm and timing. The Kodalymethod incorporates a rhythm syllables system in which note values are assigned specific syllables, whichliterally express their durations. For example, quarter notes are expressed by the syllable “ta” while eighth notepairs are expressed using the naturally shorter syllables “ti-ti.” Larger note values are expressed by extendingta to become ta-a for the half note and ta-a-a-a for the whole note (see Wheeler, 1985).


that could fill a measure in four fourths time. Second, students were taught to connect thefraction symbol with the music note (see Fig. 1). Third, students were taught to add andsubtract the fraction quantities, often with unequal denominators, represented by differentnotes to create measures of four fourths (see Fig. 2). Fourth, students were introduced toother representations of fractional quantities (i.e., fraction circles, fraction tiles, and thenumber line), taught to compare music notation and fraction symbols, and move comfortablybetween these representations (see Fig. 3). Fifth, students were taught to add and subtractfractional quantities written as a number sentence by using a representation of choice (e.g.,music note and number line) as a conceptual guide.

During the first six lessons, the students received authentic music instruction, creatingnatural musical rhythms instead of just filling measures to add up to the value of the wholenote. Rhythm relates to the relative time durations of notes and in most music a note is notsounded for more than a second (David, 1995). Rhythms in music are like the naturalrhythms in our bodies like breathing and heartbeats and relate naturally to motion. David Jr.

Table 1 Academic Music instruction

Lesson Student objectives

1 To recognize and understand whole, half, and quarter notes, their temporal size (number and lengthof 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 drummingmusic 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 quantitiesusing 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 ownform of representation if needed

12 To demonstrate the understanding of fraction equivalence by correctly adding and subtractingfractions with unlike denominators

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(1995) describes rhythm as movement in time and motion as movement in space. Thestandard of measurement in musical time is the beat and beats are generally grouped into setsof 2, 3, or 4 called measures or bars. We designated four beats in each measure with a timesignature of four fourths time. The first two lessons followed the same sequence. The musicteacher began by demonstrating how to step, clap and/or drum four lines of music, eachcontaining four measures of note patterns. First, students learned the whole note by holding aclap and taking four steps of equal value. The gesture of holding the clap, counting 4 s, andtaking four steps conveys quantifying information about the designated whole from whichwe could create other rhythms. Holding the clap for 4 s signifies the amount of time thatcreates a whole note and taking four steps presents additional information about the length orsize 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 thevaried amounts of time and length inherent in each note in our time signature as they relate toa fraction of the whole. In addition, through rhythm, students experience the note’s fractionalvalue of the whole as continuous quantities that would later help the move to a number linerepresentation. As instruction progressed, the music teacher continued to demonstrate howto 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 thepatterns 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 fractionconcepts. During this time, students were learning to add and subtract the proportionalvalues of music notes to complete measures of music. They practiced arranging variedrepeated patterns of notes (ostinatos) in each measure of four fourths time (e.g., one wholenote, two half notes, one half note and two quarter notes). We used terms that conveyed theproportional 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 introductionof 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 ofeach fraction to increase conceptual understanding (Miura, Okamoto, Vlahovic-Stetic, Kim& Han, 1999; Paik & Mix, 2003). We utilized a semiotic framework and the concrete-

Fig. 1 Sample of student moving between fraction bars, music notes, and fraction symbols


pictorial-abstract approach (CPA; Bruner, 1977) to design the elements that comprised ourinstruction. 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. Inthis way, the university researcher was able to guide students toward connecting musicinstruction to more formal mathematics language, representations, and symbols. Theselessons focused on explaining and demonstrating the concepts of fraction size and equiva-lence utilizing the connections students had made with music notes and their correspondingproportional values. Continuing the CPA sequence (Bruner, 1977), we presented repeated

Fig. 2 Sample of student moving between the number line, music notes and fraction symbols, and sample ofstudent using music notes to add fractions with unlike denominators

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note patterns in measures of four fourths time, paying careful attention to maintain themelodic quality of the patterns. First, students were taught the fraction name and symbol thatcorresponded with each note. As students began to connect the fraction name to the note andfraction symbol, the instructors assisted students with making connections between repre-sentations and moving them toward more abstract, formal fraction symbols. At the end ofeach lesson, students completed a daily worksheet including such tasks as writing thefraction symbol that corresponds to each note in several measures of music and adding theirvalues to be sure each measure contained four beats (see Fig. 2).

Starting with the seventh session, fraction circles, fraction bars, and the number line wereintroduced in succession so that students had an opportunity to partition varied representa-tions of wholes into equal parts. The compare-and-contrast structure was used to helpstudents identify the similarities and differences between partitioning different representa-tions 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 tothe corresponding portions of each representation. For example, eight eighth notes could fitbetween zero and one on the number line and in each measure of music. After comparisonswere demonstrated, students completed their own representations of fraction bars andnumber lines using notes and fraction symbols. Finally, using fraction bars, the number lineand notes, we taught students to “trade in” notes or “break up” the larger notes or fractionvalues into the smallest quantity that was in the computation problem. In this way, we taughtaddition and subtraction of fractions with unlike denominators. By focusing on the symbolicrepresentation of fractions in musical rhythms, we never referenced or taught the proceduralalgorithm (e.g., cross-multiplication) for converting unlike denominators. Students complet-ed fraction symbol problems with the option to use a chosen representational figure (i.e.,

Fig. 3 Sample of student moving between circles, music notes, and fraction symbols


music notes, fraction bars or number line). In both experimental classrooms, the regularclassroom teachers remained in the classroom for the duration of the academic musicsessions to observe and manage behavior problems if necessary. There were no behaviorproblems. However, on three occasions, the teacher in the Spanish-speaking classroomclarified directions for the class by translating them to Spanish. Outside of these occasions,we administered all sessions in English. Throughout instruction, we used transparentfraction language that described the quantity along with the name of each fraction to increaseconceptual understanding (Miura et al., 1999; Paik & Mix, 2003). Embedded throughout thelessons were phrases such as “half, one of two equal parts” and “quarter, one of four equalparts.”

Utilizing components of the Kodaly method of music instruction, students were taughtthe temporal value of the whole, half, quarter, eighth, and sixteenth notes. The whole notewas introduced as our invariant whole, which was the length of four beats. The other noteswere introduced as portions of the whole. First, students learned the whole note by holding aclap for 4 steps of equal value. Then, as new notes were introduced, students first clappedand stepped, then drummed the notes’ time values. Students learned to arrange variedrepeated patterns of notes (ostinatos) in each measure of four fourths time (e.g., one wholenote, two half notes, one half note and two quarter notes). We used terms that conveyed theproportional 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 introductionof fraction language and symbols during the following six lessons.

Each lesson followed the same instructional sequence. First, we handed out drumsticksand drum pads (inexpensive mouse computer pads) and “played” the music measures bydrumming. 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 thelesson. Second, we reviewed the prior lesson and introduced the day’s objectives. Forexample, after the students were introduced to whole, half, and quarter notes, we introducedthe 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, studentswould 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 tasksin small groups and then independently. Finally, each day, classroom worksheets werecollected and examined for student understanding and accuracy. These formative assess-ments 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 curricu-lum. Students in the comparison group were taught by their regular classroom teachers forapproximately 60 min/day, 5 days/week, using the state adopted curriculum and districttexts. The text used in the district covered the California third grade math standards andfocused on comparing fractions represented by drawings or concrete materials to showequivalency. In addition, the text explains addition and subtraction of simple fractions incontext (e.g., one half of a pizza is the same amount as two fourths of another pizza that isthe same size; show that three eights is larger than one fourth) (California Department ofEducation, 1999). Teachers in the control classrooms used the math texts and relatedworkbook 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).

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Students in the experimental group were taught by their regular classroom teachers forapproximately 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 musicintervention from the researchers during regularly scheduled math class. Both conditionsreceived 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 fractioninstruction using the district textbook. A supplemental workbook that accompanied thetextbook provided teachers with classroom practice and homework worksheets for students.Classroom instruction and worksheets focused on equivalent fractions by demonstratingrepresentations of parts of a whole and parts of a set. In addition, these representations werealso used to demonstrate addition and subtraction of simple fraction. The regular classroomteachers in the academic music classes remained in the classroom as required by the schoolprincipal 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. Afidelity checklist of crucial instructional points was constructed for each lesson using thedaily lesson plan. Crucial instructional points from each part of the lesson were identifiedand described on a fidelity checklist. These instructional points corresponded with therequired teacher behaviors for each part of the lesson (e.g., review of past lesson, introduc-tion of daily objectives, music demonstration, and presentation of new material). Fidelity oftreatment was calculated by equitably sampling 20% of the tapes from each classroom.Fidelity scores were calculated based on the number of crucial instructional points coveredin the script using the following formula: number of main points covered divided by numberof total main points×1000percentage of main points covered. Fidelity of treatment in theSpanish-speaking classroom was 96% (SD00.54), and in the English-speaking classroomwas 98% (SD00.44).

5.5 Measures

Measures used in this study were a music test (see Fig. 4), developed by the authors, and afraction concepts test, an assessment informed by Niemi (1996; see Fig. 5). Each pencil andpaper measure had two equivalent, alternate forms; problems in both forms required thesame operations and presented text with the same number and length of words. Experimentaland comparison groups were administered form A at pretest and form B at posttest. In theexperimental classrooms, we also utilized daily worksheets to monitor progress. The finalfraction worksheet was administered to both comparison and experimental groups directlyafter posttest (see Fig. 6).

5.5.1 Music test

The music test included items that required students to identify music notes, matchthe fraction that corresponds to the value of the note, and add and subtract notes andfractions to maintain four fourths time in each measure (see Fig. 4). Maximum scorefor this measure was 33. Cronbach’s alpha was .89. Interscorer agreement, calculated


by dividing the number of agreements by the number of agreements plus disagree-ments multiplied by 100, was computed on 20% of pre- and posttests by twoindependent blind scorers. Interscorer agreement was 96 and 94% for pre- andposttest, respectively.

5.5.2 Fraction concepts test

The fraction concepts test included items using regional areas, line segments, and setrepresentations to check understanding of the part-whole concept of fractions and related

Fig. 4 Music test, form B

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equality as well as fractions as numbers on the number lines (Ni, 2001; Niemi, 1996; seeFig. 5). To demonstrate understanding of fraction concepts, the test required students tomove between the fraction symbol and the fraction representation (Behr, Lesh, Post &Silver, 1983; Hiebert, 1989; Ni, 2001). Maximum score was 21 with six questions providingopportunity 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.

Fig. 5 Fraction concepts test, form A


5.5.3 Fraction worksheets

The first fraction worksheets contained fraction computation problems with like denomi-nators accompanied by representations. The difficulty level of the worksheets graduallyincreased 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) andimproper fractions, and was administered to both control and experimental groups. Whileboth groups were accustomed to completing worksheets in class, they had not completedidentical worksheets prior to completing the final worksheet. The purpose of this finalworksheet was to examine whether students’ conceptual understanding of fraction sizeand equivalency transferred to an ability to compute familiar and unfamiliar fractionswithout representations for reference. There were ten problems on the worksheet with fourproblems requiring students to find a common denominator. Regarding those four problems,students received a point for identifying the correct denominator and a point for correctcomputation. The maximum score was 14. Interscorer agreement, computed on 20% of thefinal worksheets as described above, was 95%.

5.6 Data collection

The authors administered the music test, the fraction concepts test, and the last fractioncomputation worksheet to all students in class, as a group. Pretesting on the music test andthe fraction concepts test took place 1 week before the start of the first lesson. The posttest of

Fig. 6 Final fraction worksheet, administered to all students

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the music test and the fraction concepts test took place 1 week after the last lesson wascompleted. The last fraction computation worksheet was administered directly after instruc-tion and after the posttest for the comparison group. At pre- and posttest, we read aloud thedirections and sample problems for each section and provided students with time tocomplete work before progressing to the next section. We continued to the next sectionwhen all but one student appeared to be finished.

6 Results

6.1 Analysis of treatment effects

Table 2 shows students’ mean scores and standard deviations for pre- and posttest perfor-mance on the music test and the fraction concepts test. The table also shows students’ meanscores for the last fraction worksheet. Independent samples t tests comparing the pretestmean scores of the experimental and control group on the music test and the fractionconcepts test revealed no significant differences, t(65)0−.496, p0 .62 and t(65)0 .36,p0 .72, respectively.

6.1.1 Music notation knowledge

With regard to analyzing comparison versus experimental groups’ performance on the musictest, we planned to use ANCOVA and first tested the assumption of equal slopes utilizing ageneral linear model with the music posttest as the dependent variable, treatment as a fixedfactor and the music pretest as a covariate. The interaction of the music pretest and academicmusic instruction was significant, F(1, 63)08.69, p0 .004, ηp

20 .12. Due to this interaction,we did not perform an ANCOVA analysis, and instead examined the scatter plot and best-fitlines. The interaction suggested that although all experimental students benefited frominstruction, the positive effect of the treatment was higher among students who scored high

Table 2 Performance data

Variables Comparison Experimental Comparison Experimental

n030 n037 Subgroupsa Subgroupsa

X (SD) X (SD) X (SD) X (SD)

Music test n016 n016

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 n010 n015

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)*

*p<.05a Subgroub includes only those students who performed below the Grand mean on the pretest


on the pretest than students who scored low on the pretest (see Fig. 7). An independent samplest test revealed that the experimental group’s mean posttest score on the music test (M015.78,SD08.46) differed significantly from the comparison group’s mean score (M06.23, SD04.13),not assuming equal variances, t(55.5)06.04, p<.01, ES01.46 using Cohen’s d. Because wewere interested in how the academic music instruction affected students who performed belowthe mean on the music pretest, we examined the differences of comparison versus experimentalstudents’ scores for the posttest by splitting the sample at the grand mean (M03.94, SD02.56)of pretest performance scores and excluding those students who performed above the grandmean on the music pretest. From the smaller sample of students who performed below the grandmean on the music pretest, the experimental group (n016;M011.33, SD07.01) outperformedthe comparison group (n016; M05.44, SD03.85) on the music posttest, t(23.28)0−2.84,p<.01, equal variances not assumed, ES01.04 using Cohen’s d.

6.1.2 Fraction concepts test

With regard to examining comparison versus experimental group’s performance on thefraction concepts test, we planned to use ANCOVA and first tested the assumption of equalslopes, utilizing a general linear model with the fraction concepts posttest as the dependentvariable, treatment as a fixed factor and the fraction concepts pretest as a covariate. Theinteraction of the fraction concepts pretest and academic music instruction was significant,F(1, 63)07.31, p0 .009, ηp

20 .10. We instead examined the scatter plot and best-fit lines.

Fig. 7 Scatter plot of music test results

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These suggested that students from the experimental group who performed lower on thepretest appeared to benefit more from academic music instruction than their peers in theexperimental group who performed higher on the pretest (see Fig. 8). In comparing theperformance of the experimental group to the comparison group, an independent samples ttest revealed that the experimental group’s mean posttest score on the fraction concepts test(M013.43, SD02.51) did not differ significantly from the comparison group’s mean score(M012.20, SD03.09), not assuming equal variances, t(55.5)01.76, p0 .08, ES0 .44 usingCohen’s d. We again examined the differences of control versus experimental groups’ scoresfor the posttest by splitting the sample at the grand mean (M08.03, SD03.36) of pretestperformance scores and excluding those students who performed above the grand mean onthe fraction concepts pretest. From the smaller sample of students who performed below thegrand mean, experimental students (n015; M011.60, SD02.26) outperformed comparisonstudents (n010; M09.10, SD02.28) on the fraction concepts posttest, t(19.32)0−2.69,p0 .01, equal variances not assumed, ES01.15 using Cohen’s d.

6.1.3 Fraction computation

We used an independent samples t test to examine the effects of academic music instructionon fraction computation between the comparison and experimental group’s performance onthe final fraction worksheet (see Table 3). An independent samples t test revealed that theexperimental group’s mean score on the final fraction worksheet (M07.08, SD03.88)

Fig. 8 Scatter plot of fraction concepts test results, including pre- and posttest data


differed significantly from the control group’s mean score (M03.87, SD02.21), with theexperimental group outperforming the comparison group, t(58.83)01.76, p<.01, equalvariances not assumed, ES01.00 using Cohen’s d. Again, because we were interested inhow academic music instruction affected the fraction computation skills of those studentswho came to instruction with less conceptual knowledge of fractions as determined by theirpretest scores on the fraction concepts pretest, we examined the differences of comparisonversus experimental group’s scores for the final fraction worksheet by splitting the sample atthe grand mean (M08.03, SD03.36) of pretest fraction concepts scores and excluding thosestudents who performed above the grand mean. From the smaller sample of students whoperformed below the grand mean, the experimental group (n015; M05.67, SD04.79)outperformed the comparison group (n010; M03.30, SD02.79) on the final fraction work-sheet, t(22.72)0−1.56, p0 .02, equal variances not assumed, ES0 .60 using Cohen’s d.

On an error analysis of the fraction worksheet, students in the experimental group and thecomparison group showed varying patterns of errors. As shown by Table 4, errors in theexperimental group due to mistakes in calculating a common denominator (21%) were morelikely than in the comparison group (5%). On the other hand, errors in the experimentalgroup were less likely to be due to faulty procedural application (8%), such as adding orsubtracting across the numerator and denominator, than errors in the comparisongroup (24%). Students in the experimental group were slightly less likely to leave aproblem blank that required multiplication or conversion to a common denominator(11%) than students in the comparison group (19%).

7 Discussion

By introducing a number of fraction representations in a stepwise fashion across theacademic music intervention, we created a semiotic chain (Presmeg, 2006), a sequence of

Table 3 Percentage of correct responses for computation problems by group

Entire sample Smaller samplea

Experimental Comparison Experimental Comparison

# Problem n037 n030 n015 n010

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

Note: Percentages are rounded to the nearest percent and represent the students who answered each questioncorrectly. Entries in italics refer to transfer questions containing fractions not covered in the academic musicinterventiona Subgroup includes only those students who performed below the grand mean on the fraction concepts pretest

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abstractions that started with music notation (the first signs) and then introduced discrete andcontinuous representations of fractions (the second signs). We completed the chain withformal fraction symbols (the third signs). In this way, we developed students’ understandingof fraction size and equivalence in a series of steps that started with musical activities andprogressed toward more formal mathematical concepts of fraction equivalence and size. Wewill begin by addressing the research questions and the subsequent theoretical underpinningsthat explain our results. Second, we will discuss the social validity of the academic musicintervention, study limitations, and future research possibilities. Finally, we will summarizeour 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 testsuggests that the experimental group acquired significantly more music notation knowledgethan the comparison group, who did not receive this type of instruction. In fact, the meanscore of students in the experimental group was almost one and a half standard deviationsover the mean score of students in the comparison group at posttest. This disparity mostlikely reflects the design of the study in which the comparison group only receives standardfraction instruction while the experimental group had an additional music component.

Academic music required students to demonstrate their understanding of the value ofnotes with clapping, drumming, reciting names, and adding and subtracting the values ofnotes to complete measures. The music notes acted as the first signs or representation in thesemiotic chain leading to more formal mathematic symbols. The instructional gesturesserved as signs or representations for fractional temporal units and provided repeated

Table 4 Error analysis

Error type Experimentalgroup (n037)

% oferrors

Comparisongroup (n030)

% oferrors

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 andfraction (2× 1/2) or multiplied denominator only/incorrectmultiplication

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%

Percentages are rounded to the nearest percent and represent the error type by total errors possible in eachgroup


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 tofractions symbols, students may have already internalized their understanding of size dueto the experience of drumming the temporal values of the notes. It may have been the dailyacts of drumming and saying each note’s name in Kodaly syllables that provided akinesthetic, oral, and visual mode for experiencing temporal values of notes and relatedfractions. The gesture of students holding a clap or holding the drumstick still for four countsand saying ta-ah-ah-ah as a representation of the whole note provided opportunity for themto experience the value of the whole. Notes of shorter duration, like the half note and quarternote, provided opportunity for students to experience more claps or drums with notes ofshorter duration. We suggest that this experience helped students to more readily understandthat the smaller the denominator (note value), the bigger the numerator (number of notes) toequal the value of the whole. In turn, holding the clap or drumstick for four counts andbreaking the measure up with several smaller notes provided a natural transition to using thenumber line when representing fractions.

7.1.2 Transfer of fraction understanding

Moving between representations of fractional quantities and fraction symbols is one of themost difficult aspects of fraction instruction (Hiebert, 1989; Mack, 1995). However, utilizingphysical gestures and various representations associated with fractional quantities appearedto help struggling students as they worked to identify, add, and subtract notes and fractionalquantities to create measures in four fourths time (see Figs. 1, 2, and 3). Bruner emphasizedthe use of three distinct kinds of materials in teaching mathematics to children that tie in tothe CPA sequence: enactive, iconic, and symbolic (Ediger, 1999). Enactive materialsaccentuate the use of concrete models for children to learn using a hands-on approachthrough manipulation of the materials. Iconic materials correspond to pictoral representa-tions and highlight the use of pictures, illustrations, and other visual aids. Symbolic materialsare more abstract and emphasize the use of textbooks and more formal mathematicalcontent. The underlying idea is to begin instruction using concrete models and moveprogressively through more abstract ones, helping students build strong connections alongthe way.

Our concrete level of instruction involved clapping, stepping and drumming the value ofmusic notes, using hands and drumsticks as manipulatives. The pictorial level involvedstudents appropriately filling music measures in four fourths time with varied patterns ofnotes or conversely removing notes to maintain the four fourths time within each measure. Inaddition, the value of notes was alternately compared to parts of fraction circles, fraction barsand places on the number line. The abstract level of instruction involved moving studentsaway from musical notes and fractional representations to using only fraction symbols toadd, subtract, and multiply fraction quantities.

The music test (see Fig. 4) and fraction concept test (see Fig. 5) measured transfer offraction reasoning between music notation and fraction symbols. Overall, students in theexperimental group significantly outperformed students in the comparison group on themusic test. Students in the experimental group demonstrated the ability to transfer knowl-edge of music notes to fraction symbols. They were also able to add and subtract propor-tional values of the notes and fraction symbols to create measures that fit into the keysignature. The ability to create measures in four fourths time with both notes and fractionsymbols showed that these students were gaining an emerging understanding of fractionreasoning. As measured by the fraction concepts test, both students in the experimental and

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comparison groups made significant improvements in their conceptual understanding ofbasic fractions. Because all third grade students were being exposed to introductory fractionconcepts in the standard curriculum, the mean scores of all students improved from pre- toposttest. While we found no significant mean difference between the students in theexperimental group and the comparison group on the fraction concepts posttest, examinationof the scatterplots suggests two explanations. First, this test aligns more closely with thestandard curriculum, so all students taught through that curriculum would be expected tomake similar progress. Second, students in the experimental group who came to academicmusic with less conceptual knowledge than their lower achieving peers in the comparisongroup (based on pretest fraction concept scores) may have utilized their emerging fractionreasoning knowledge to outperform their peers in the comparison group at posttest. Inaddition, students who performed below the grand mean in the experimental group scoredover one standard deviation higher than their peers in the comparison group. While thesample is small and we must be careful not to overgeneralize, the students who came toinstruction with less conceptual knowledge at pretest appeared to gain significantly from thisinstruction. Lower performing students often require more intensive interventions thanwhole class instruction to show evidence of emerging understanding, so we are encouragedby their positive response to academic music.

7.1.3 Fraction computation

Students in the experimental group outperformed students in the comparison group on thefinal fraction worksheet with a strong effect size. The mean score for students in theexperimental group was about one standard deviation higher than the mean score of studentsin the comparison group. Students in the experimental group clearly outperformed theirpeers in the comparison group on every item, but there are important differences in the typesof errors made. For example, students in the experimental group were far less likely to addacross numerator and denominator, a common mistake made by students learning to add andsubtract fractions, which often persists into high school (Newton, 2008). Twenty-fourpercent of total errors made by the comparison group compared to 8% of total errors madeby the experimental group were due to adding or subtracting across numerator and denom-inator. Mack (1990) suggests that teaching formal algorithms and rote procedures beforestudents have developed conceptual understanding of basic fraction concepts may causestudents to blindly apply procedures. Because only 8% of the total errors made by studentsin the experimental group were due to adding or subtracting the numerator and the denom-inator, those students may have utilized the connections they made between the fractionalvalue of notes and equivalent fraction symbols to inform their computation. When adding orsubtracting fractions with unlike denominators, students in the experimental group averagedabout 28% correct while students in the comparison group averaged about 3% correct.

Students in the experimental group may have utilized their conceptual understanding of theproportional sizes of music notes and related fraction symbols to inform computation ofunfamiliar fractions. By contrast, students in the comparison group may have relied onprocedural knowledge to calculate without fully understanding proportions. These explanationsare based on an error analysis of the final fraction worksheet (Table 4). This worksheet includeditems 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 itemthat required conversion or multiplication blank was 19% by students in the comparison groupversus 11% by students in the experimental group. Students in the experimental group seemedto 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 mathematicsquestions, they may gain confidence, lessen their mathematics anxiety, and improve theirperformance (Ashcraft & Krause, 2007; Furner & Berman, 2004; Shields, 2005).

All students in this study were taught a procedural algorithm for computing fractions withunlike denominators in their regular math class. However, the comparison group may haverelied only on the algorithm but they did not use it consistently. Though the comparisongroup’s performance on the fraction concepts test was comparable to the experimentalgroup’s performance, it may have been that the comparison group was overly reliant onthe learned procedure for computing with unlike denominators and less likely to utilize theirconceptual knowledge to inform computation. While the experimental group demonstratedan understanding of the need to convert unlike denominators before adding or subtractingfractions, 21% of their total errors were due to incorrect conversions (e.g., trying to convertsmaller denominators to larger denominators for addition or subtraction). In examining theactual test protocols, diagrams and calculations suggest that students in the experimentalgroup knew a conversion was required to compute unlike denominators but those individ-uals with more fragile understanding either drew notes larger than should be used or theymade addition or subtraction errors. By contrast, fewer errors in the comparison group weredue to incorrect conversions (5%) because no attempt at conversion was made. Students inthe comparison group appeared less aware of the need to convert unlike denominators.Students in the experimental group had more opportunities to practice proportional reason-ing with unlike denominators during academic music instruction. So, even though somefractions had unfamiliar denominators (e.g., one third), they were more likely to attempt totransfer fraction reasoning to novel problems. The relative inability of the comparison groupto transfer knowledge to novel problems suggests a more fragile conceptual understandingof fraction size and equivalency, and demonstrates the importance of providing more ampleopportunities for students to solve novel and difficult fraction problems. However, therelatively low performance of all students on the transfer questions shows that the develop-ment of fraction skills can be slow and complex. With practice, students can develop theexperience necessary to correctly apply conceptual knowledge and procedural strategies tomore complex fraction problems (Brigham et al., 1996).

It appears that without a representational picture to show the size of the units involved in theproblem, some students may have allowed their whole number knowledge to interfere withchoosing the correct, or smallest, denominator to divide into equivalent parts for calculation.Mack (1990) found that when students are first introduced to fractions, they frequentlyconfound whole number principles with newly developing fraction concepts, especially whenthey 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 proceduralalgorithm 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 theexperimental group. Also, the daily tasks of drumming to make the music sounds and thecognitive movement between a variety of representations in the experimental group may haveprovided adequate opportunities for students to begin constructing meaning for the size andequivalency 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- toposttest on the music test and the fraction concepts test. Students in the comparison group

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also made significant gains from pre- to posttest on the fraction concepts test as the test wasaligned with the curriculum in place at the school. However, experimental students whocame to instruction with less conceptual knowledge than their higher achieving peers madesignificant 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 whocame to the intervention with less conceptual knowledge appeared to more readily constructan understanding of fraction concepts and surpassed similar students in the comparisongroup. Students in the experimental group appeared to benefit from the addition of academicmusic instruction to nearly meet the mean performance of their peers and also madesignificant gains over students in the comparison group at posttest. In the experimentalgroup, students with pretest scores below the grand mean scored posttest scores over onestandard 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 signif-icant gains on the Conceptual Knowledge test, those lower performing students in the groupdid not gain as much as the lower performing students exposed to academic music. Oneexplanation may be that lower performing students benefit from the multimodal semioticapproach 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 oftreatment, procedures, and the outcomes produced by these procedures (Foster & Mash,1999). The goal of academic music was to teach fraction concepts through authentic musicinstruction. In this way, we addressed an important instructional hurdle in the elementarycurriculum, fraction concepts, and simultaneously provided music instruction, which is anoverlooked core content subject in the elementary curriculum. Our procedures includedevidence-based instructional practices that are effective for all students, especially studentswith learning differences (Bruner, 1977; Ediger, 1999; Miller & Hudson, 2007).

Our procedures were clearly outlined in lesson plans and could be put together withinstructions to create an easy-to-use manual. Both experimental teachers said they would andcould use academic music in their classrooms with some training. The outcomes producedby our academic music procedures suggest these students gained a better conceptualunderstanding of fraction size and equivalency than students who spent all of their mathinstructional time learning from the standard curriculum. In addition, academic musicbrought music back into the classroom with genuine music instruction, which is lacking inmany underfunded schools. We suggest that using music instruction to indirectly teachfraction concepts may serve as an important motivator for students to learn and persist inunderstanding difficult fraction concepts. In addition, teachers’ willingness to teach aca-demic music suggests the program may address teachers’ discomfort when teaching difficultmathematics concepts (Frykholm, 2004) and the lack of conceptual instructional guidance inclassroom 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 twocomparison classrooms from the same school for a total of 67 participants. Second, studentswere not randomly assigned to comparison and experimental group, but assigned byclassroom to conditions by the school principal. Third, approximately half of the studentsin 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 samplecharacteristics, results may not occur with a different sample. Finally, while both teachers inthe academic music group thought they could implement academic music in their class-rooms, we cannot be certain that they would obtain similar results if they did not have somemusic training. Though we include specific instructions in lesson plans and worksheets, andthe basis of the music component is rhythm, it is nonetheless important that the sound of themeasures 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 toteach fraction concepts in the elementary curriculum. We have a compelling reason to viewmusic instruction as an integral part of the elementary curriculum, due to its utility inteaching beginning fraction concepts and related fraction computation to elementary stu-dents. Furthermore, this intervention appears to be particularly effective for students who arecoming to instruction with a lower than average understanding of fractions. Academic musicappears to have strengthened their conceptual understanding of both the magnitude andequivalency of fractions via a semiotic game. Curriculum developers must keep in mindways to address learners who do not initially respond to sound classroom instruction, withthe goal to reach all learners.

Radford, Edwards and Arzarello (2009) describe in a special issue of Educational Studies inMathematics how semiotic instruction can be effective for students at varied levels of under-standing because it helps students make connections between multiple representations of impor-tant ideas and encourages interactions with ideas and expressions in new ways. In a more recentspecial issue on semiotics, Radford, Schubring and Seeger (2011) further describe how teachingand making meaning of difficult mathematics concepts have been influenced by this blossomingsemiotic perspective. The authors suggest that meaning making can be a communal classroomactivity system in which teachers “create zones of proximal development where students makemeaning” (Radford et al., 2011, p. 155). In this way, semiotic instruction, using multimodalresources and gestures, may help increase conceptual understanding of fractions and proportions.Academicmusic 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-gradestudents with and without learning differences. As we focused on fraction symbols, fractionsize and fraction equivalency, we taught students that when ordering fractions or findingequivalent fractions, the whole must be the same size. While teaching the invariance of thewhole may have enabled students to more readily learn these beginning concepts, it mayimpede instruction later when students are confronted with a variable unit in problemsolving. Because we utilized several representations of fractions, future instruction thatintroduces the variable unit can continue with several of these already familiar representa-tions. For example, the number line is particularly well suited to introduce a variable unitbecause a length can represent a unit and the number line model allows for iteration of theunit 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 feelthe need to point out that future instruction must address the notion that a fraction representsa relative amount.

Future research will include creating an academic music manual, with a CD or DVD withmusic examples, complete with lesson plans and instructions for teachers to use in their own

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classrooms. We hope that teachers can administer academic music to their students withintegrity and fidelity to achieve similar results. Future studies will examine the effects of amore general program, rooted in the concept of semiotics, to increase global mathematicsachievement of students in elementary school.

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