1 does homework design matter; the role of homework's purpose in student mathematics achievement

7/25/2019 1 Does Homework Design Matter; The Role of Homework's Purpose in Student Mathematics Achievement

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Does homework design matter? The role of homeworks purpose instudent mathematics achievementPedro Rosrioa,*,Jos Carlos Nez b, Guillermo Vallejo b, Jennifer Cunhaa, Tnia Nunes a,Rosa Mouroa, Ricardo Pinto a

a Department of Applied Psychology, Universidade do Minho, Braga, Portugalb Department of Psychology, Universidad de Oviedo, Oviedo, Spain

A R T I C L E I N F O

Article history:

Available online 13 August 2015

Keywords:

Homework purposesMathematics teachersAmount of homeworkRandomized pretestposttest clustereddesign

A B S T R A C T

This study used a randomized pretestposttest clustered design to examine the effect of 3 homeworkpurposes (i.e., practice, preparation, and extension) on 6th graders mathematics achievement and howthis relationship was modulated by the amount of completed homework. A total of 27 mathematics teach-ers and their 638 students participated in this study. Once a week for six weeks, the teachers assignedtasks that had a specific type of homework purpose according to their treatment condition. At the endof the six weeks, the students completed a non-standardized mathematics achievement test. The resultsof multilevel modeling showed that after controlling for student characteristics and class-level vari-ables, extension homework positively impacted students mathematics achievement, while practice andpreparation homework did not. These findings were not related to the amount of homework that wascompleted by the students. The findings highlighted the importance of the teachers role in the first phaseof the homework process (i.e., designing homework with a specific purpose) and provide important datafor teachers and school administrators to reflect upon when conducting actual homework practices.

2015 Elsevier Inc. All rights reserved.

1. Introduction

The Russians launch of the Sputnik satellite in the late 1950sprovided an opportunity to reflect on the quality of educationalsystems, specifically the American system, for preparing new gen-erations for a technological and competitive world (Cooper, 1989;Cooper, Robinson, & Patall, 2006). Homework policies were amongthe American educational systems response for addressing theseeducational challenges(Cooper, 1989; Cooper et al., 2006). In theliterature, homework is defined as the set of school tasks that areassigned by teachers forstudents to complete outside of school hours(Cooper, Steenbergen-Hu, & Dent, 2012) and viewed as an impor-tant instructional tool (Cooper & Valentine, 2001; Fernndez-Alonso,Surez-lvarez, & Muiz, 2015; Hagger, Sultan, Hardcastle, &

Chatzisarantis, 2015; Marzano & Pickering, 2007). The homeworkprocess is complex and involves three actors (i.e., students, parentsand teachers), who have goals and behaviors that can sometimesbe misaligned and in conflict (e.g., Cooper, 1989; Cooper et al., 2006;Nez et al., 2015; Trautwein & Kller, 2003; Warton, 2001). Pre-vious research has primarily focused on students perceptions ofhomework behaviors (e.g., homework management, homework

effort, homework completion, homework emotions) (Dettmers et al.,2011; Goetz et al., 2012; Rosrio et al., 2011; Trautwein, Ldtke,Schnyder, & Niggli, 2006; Xu, 2011; Xu & Wu, 2013) and parentalinvolvement in homework (e.g., Dumont, Trautwein, Nagy, &Nagengast, 2014; Patall, Cooper, & Robinson, 2008). Despite the im-portance of teachers variables on the homework process, there islimited research on the impact of teachers homework practices onstudents learning and achievement (e.g., Bang, 2012; Epstein & VanVoorhis, 2012; Trautwein, Niggli, Schnyder, & Ldke, 2009). Thus,the present study aims to fill this research gap with a study of theimpact of homework purposes on students academic achievement.

1.1. Factors that influence homework

The Cooper homework model(1989) synthetizes factors that havethe potential to influence the effect of homework on students home-work behaviors and academic achievement (e.g., exogenous factors,such as student characteristics, assignment characteristics, initialclassroom factors, home-community factors, and classroomfollow-up).

Due to limited space, we only address the variables that encom-pass homework assignment characteristics. According toCooper(1989),homework assignments can vary in the following aspects:the amount (empirically measured by the amount of time stu-dents spend completing homework and the frequency students areassigned homework), purpose, degree of individualization, degree

* Corresponding author. Escola de Psicologia, Universidade do Minho, Gualtar,4710-052 Braga, Portugal. Fax: 25604224.

E-mail address:[email protected](P. Rosrio).

http://dx.doi.org/10.1016/j.cedpsych.2015.08.001

0361-476X/ 2015 Elsevier Inc. All rights reserved.

Contemporary Educational Psychology 43 (2015) 1024

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Contemporary Educational Psychology

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of choice, completion deadlines, and social context (i.e., the way thatstudents complete the homeworkindividually, in a group, askingfor help).

Recently, research on homework has analyzed the relationshipbetween several of the previously mentioned homework vari-ables, students homework behaviors and their relationship toacademic achievement (e.g.,Cooper, Lindsay, Nye, & Greathouse,1998; Cooper et al., 2006; Nez, Surez, Cerezo, Rosrio, & Valle,2013; Trautwein, Kller, Schmitz, & Baumert, 2002; Xu, 2008). Theamount of time spent on homework is one variable that has at-tracted attention from researchers (e.g.,Cooper et al., 2006; Keith,1982; Trautwein & Ldtke, 2007; Trautwein, Schnyder, Niggli,Neumann, & Ldtke, 2009). There have been mixed results for therelationship between the amount of time spent on homework andhomework effectiveness (e.g.,Dettmers, Trautwein, & Ldtke, 2009;Nez et al., 2013; Trautwein, 2007). For example,Keith (1982)re-ported a positive association between the amount of time spent onhomework and studentsacademic achievement in high school, whileother authors found low, null or negative relationships in elemen-tary, middle and high schools (e.g.,Cooper et al., 2006; Nez et al.,2013; Rosrio et al., 2009, 2011; Trautwein, Schnyder et al., 2009).In contrast, research that has examined homework frequency hasconsistently shown a positive association between homework fre-

quency and academic achievement (e.g.,Coleman, Hoffer, & Kilgore,1982; Dettmers, Trautwein, Ldtke, Kunter, & Baumert, 2010; Farrow,Tymms, & Henderson, 1999; Fernndez-Alonso et al., 2015).

There are extensive data on the relationship between home-work behaviors (e.g., the amount of time spent on homework andhomework frequency) and students academic achievement;however, there is limited research on the relationship between dif-ferent types of homework purposes and academic achievement(Epstein & Van Voorhis, 2001, 2012; Hallam, 2004; Warton, 2001).In fact, the phase of homework preparation (e.g., the design andpurpose of the assigned tasks) has not yet been extensively studieddespite its importance to the subsequent steps in the homeworkprocess (e.g., Bang, 2012; Epstein & Van Voorhis, 2001, 2012; Warton,2001).

1.1.1. The role of homework purposes

Despite the relationship between homework behaviors and stu-dents academic achievement, assigning more homework does notlead to better homework performance when teachers do not con-sider other homework characteristics, specifically the purpose foreach homework task (Epstein & Van Voorhis, 2001).Lee and Pruitt(1979)proposed a description of homework assignment purposesto increase the benefits of homework tasks. These authors de-scribed four types of instructional homework purposes: practice (i.e.,practicing the material that is covered in class to master skills), prep-aration (i.e., preparing the next lesson), extension (i.e., transferringprior learning to new situations) and creative (i.e., integrating severalcompetencies into one task as a research project) (Lee & Pruitt, 1979,

p. 32). The same authors call for teachers to attend to the impor-tance of assigning homework tasks that are aligned with purposesthat can promote students engagement and meaningful learning.

Epstein and colleagues research on the topic (see Epstein & VanVoorhis, 2001; Van Voorhis, 2004)identified 10 homework pur-poses that can be organized into three groups: instructional (i.e.,practice, preparation, participation, and personal development),communicative (i.e., parentchild relations, parentteacher com-munication, and peer interactions), and political (i.e., policy, publicrelations and punishment). Recently, Epstein and VanVoorhis (2012)reinforced that homeworkpurposes are an essentialaspect of hom-eworks ability to maximize impact on students learning andacademic success. Accordingto these authors, when homeworktasksare devoid of clear homework purposes, students are more likely

not to complete the homework. Thus, as Epstein and Van Voorhis

(2012) suggested, there is a call to clarify the effects of homeworkscontrasting purposes on students academic results. These findingscould provide information on teachers and school administratorshomework practices, schools homework policies and, consequent-ly, improve homework practices.

Based on previous research (e.g., Epstein, 1988, 2001; Epstein& Van Voorhis, 2001; Lee & Pruitt, 1979),Cooper (2001)definedhomework purposes as an assignment characteristic that could po-tentially influence homeworks utility. He proposed four instructionalhomework purposes, as follows: practice or review, preparation, ex-tension and integration (for a full description, see Cooper, 2001).Moreover, Cooper (2001) stated that homework can serve other pur-poses that are not related to instruction: parentchild or parentschool communications, directives from school administrators andstudent punishments. Several years later,Cooper et al. (2006)con-cluded that practicing and reviewing the material taught in classwas the most frequent homework purpose used by teachers. Otherstudies that were conducted in different academic domains (e.g.,mathematics, language, physics, science) and with different gradelevels (i.e., elementary and middle school) support these findings(e.g., Danielson, Strom, & Kramer, 2011; Kaur, 2011; Tas,Sungur-Vural, & ztekin, 2014). However, these studies did notaddress the impact of homework purposes on students academic

achievement.More recently,Xu (2008)proposed a taxonomy that comprised

fifteen homework purposes that were then reduced to three (i.e.,peer-oriented, adult-oriented and learning-oriented reasons) as aresult of a factor analysis that assessed the validity of the home-work purposes scale(Xu, 2010a). In a multilevel study,Xu (2010b)analyzed the relationship between 8th- and 11th-grade studentsperceptions of homework purposes and found several variables thatwere related to the homework process (i.e., interest in homework,feedback perceived by students, affective attitude toward home-work and homework help from the family). The author found thatthe three homework purposes (i.e., peer-oriented, adult-orientedand learning-oriented reasons) were positively associated with af-fective attitude toward homework, interest in homework, teacher

feedback and homework help from the family. The studies byXu(2008, 2010a, 2010b)focused on students perceptions and did notexamine teachers reports on the purpose of the homework. More-over, the relationship between homework purpose and academicachievement was not addressed in Xus studies (2008, 2010a, 2010b).

To our knowledge, only two studies have analyzed the influ-ence of homework purposes on students academic achievement,with conflicting results (i.e., Foyle, Lyman, Tompkins, Perne, & Foyle,1990; Trautwein, Niggli et al., 2009).

Foyle et al. (1990)used an experimental design in a classroomwith 64 5th-grade social studies students. They found that home-work purposes (i.e., homework for preparation and practice)combined with cooperative learning improved 5th graders socialstudies achievement compared to a group with no assigned home-

work. The authors concluded that students in the group withassigned homework increased the amount of time spent on task andbenefitted from the twotypes of homework assignments (Foyleet al.,1990). However, there were no significant differences between thetwo types of homework purposes (i.e., practice and preparation).The authors suggested that there is a need for additional researchto verify whether their findings could be generalized to other sub-

jects. In contrast, Trautwein, Niggliet al. (2009) used teachers reportsof 8th-grade French as a second language students to analyze thepredictive effects of homework objectives (including homework pur-poses), other variables that were related to teachers practices andbeliefs toward homework, and students homework effort, emo-tions and academic achievement. The results indicated that studentshad lower grades in classes that assigned homework tasks with drill

and practice objectives compared to classes that had homework

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assignments that were designed for other purposes (e.g., to promotestudents motivation). Using multilevel modeling, this study con-tributed new insight into the role of homework purposes on studentoutcomes; however, homework purposes were assessed by teach-ers reports rather than homework purpose practices. Moreover, thisstudy did not address the issue of causation; all variables were col-lected at the same time, and, thus, the authors suggested that futurestudies should use a longitudinal approach. Finally, Trautwein, Niggliet al. (2009)suggested that future research should analyze the re-lationships between these variables in research in other academicdomains and grade levels.

2. The present study

The current study was designed to examine the relationshipbetween three instructional homework purposes (Cooper, 2001;Epstein & Van Voorhis, 2001; Lee & Pruitt, 1979)and mathemat-ics achievement in 6th-grade students using multilevel modeling.Multilevel models provide an innovative method for examininghomework purposes in mathematics in this quasi-experimentalstudy.

To our knowledge, research on homework purposes has been con-

ducted on French as a second language (Trautwein, Niggli et al., 2009)and social studies (Foyle et al., 1990); the opposing results couldbe due to the different school subjects or grade levels that were usedin the two studies. We chose to study mathematics because it hashigh educational relevance with other subjects (e.g., science, tech-nology, and engineering) and professional development (e.g., seeHagger et al., 2015; OECD, 2013). Moreover, because mathematicsis a school subject that has a high use of homework (e.g., Rnning,2011; Xu, 2015), it is important to understand how to improve hom-eworks effectiveness on students learning and academicachievement.

Moreover, because Portuguesestudents scores on the PISA 2012were above the OECD average (OECD, 2014a), examining the rela-tionshipbetweenhomework purposesand mathematics achievement

could help to design good practices for mathematics homework, inaddition to teacher practices. Importantly, interviews with schooladministrators and mathematics department directors revealed thatthere are important differences in expectations between studentsat the end of elementary school (i.e., the 6th-grade level) and typesof homework purpose (i.e., practice, extension).

The current research only focuses on instructional purposesbecause the literature indicates that homework assignments withthis purpose are more related to the content of achievement testsand, therefore, to students academic achievement (Cooper et al.,2006; Muhlenbruck, Cooper, Nye, & Lindsay, 2000) compared to non-instructional purposes (e.g., to promote communication betweenhome and school). Furthermore, instructional homework pur-poses are most often assigned by teachers(Cooper et al., 2006;

Danielson et al., 2011; Foyle et al., 1990; Kaur, 2011; Tas et al., 2014;Trautwein, Niggli et al., 2009).Previous research that has used non-experimental designs sug-

gests the need for conducting controlled studies to better addressthe issue of causation (e.g., Dettmers et al., 2011; Ramdass &Zimmerman, 2011; Xu, 2011; Xu & Wu, 2013). Although the abilityto control for all variables is limited in the classroom context, thepresent study uses a randomized pretestposttest clustered designso that homework purposes are as ecologically valid as possible inan authentic learning environment.

Thepresent study controlled forgender,prior knowledgeof math-ematics, mathematics self-efficacy, amount of time studyingmathematics, and parents educational level. Recent research sug-gests that, compared to boys, girls organize their work space better

and more efficiently monitor their motivational processes, for

example, controllingthe negative emotions thatemerge while doinghomework (Xu, 2010a). First, compared to boys, girls report usingthese strategies to morefrequently complete homework (Xu, 2007),spend more time doing their homework (Trautwein, 2007; Wagner,Schober,& Spiel, 2007) and have more positive attitudes toward thehomework(Rosrio,Mouro,Nez, Gonzlez-Pienda,& Valle, 2006).Thus, girls are less likely to attend class without having completedtheir homework and are more disposed to view homework as lessboring than boys(Xu, 2006). Second, prior knowledge is an impor-tant variable that influences the relationship between homeworkand academic achievement(Trautwein et al., 2002;Trautwein,Schnyder et al., 2009). Data from several studies showed that priorachievement positively and significantly predicted students aca-demicachievement (e.g., Fernndez-Alonso et al.,2015; Hemmings,Grootenboer, & Kay, 2011; Nez, Vallejo, Rosrio, Tuero, & Valle,2014; Zuffian et al., 2012). Third, self-efficacy was controlled forto avoid biased results because it is a positive predictor of academ-ic achievement in elementary (e.g.,Rosrio et al., 2009, 2011) andsecondary schools (e.g.,Kitsantas, Cheema, & Ware, 2011; Lee, Lee,& Bong, 2014). Moreover, the amount of time spent studying is pos-itively associated with academic achievement (e.g.,Carroll, 1963;Rosrio et al., 2009, 2011). Finally, the literature has shown that theparents level of education has a positive impact on students aca-

demicperformance (e.g., Fernndez-Alonso et al.,2015; Gustafsson,Hansen, & Rosn, 2013; Nez, Vallejo et al., 2014;OECD, 2013;Rosrio et al., 2005). Although parents vary in their educational in-tentions, strategies,and actions, they generally believe that helpingtheir children with their homework is a parental responsibility(Epstein & Van Voorhis, 2012; Hoover-Dempsey, Bassler, & Burow,1995).

To address the call for research on the role of homework pur-poses (one of the homework characteristics) (Epstein & Van Voorhis,2012; Foyle et al., 1990; Trautwein & Ldtke, 2007; Trautwein,Schnyder et al., 2009; Warton, 2001), we conducted a quasi-experimental design to analyze the relationship between the threeinstructional homework purposes (i.e., practice, preparation, and ex-tension; see Cooper, 2001; Epstein & Van Voorhis, 2001; Lee & Pruitt,

1979), the amount of homework assignments completed and math-ematics achievement. The purpose of our study was twofold. First,we examined the impact of teachers homework purposes on stu-dents mathematics achievement. Second, we analyzed how thisrelationship (between the type of homework purpose and mathe-matics achievement) could be modulated by the amount ofcompleted homework.

To address these goals, we conducted a randomized pretestposttest clustered design. Specifically, 27 teachers (classes) wererandomly assigned to three different homework purpose condi-tions (i.e., practice, preparation and extension), with 9 teachers ineach treatment group. Over a six-week period, once a week, eachteacher assigned homework according to the treatment condition.At the end of the six weeks, the effects of homework purpose on

mathematics achievement were assessed with an exam that coveredthe content that was worked on in class (see section 3.2).In the current study, we addressed the following specific re-

search questions:

1. Are there statistically significant differences in mathematicsachievement between the different types of teachers home-work purposes?

2. What types of teachers homework purposes are the mosteffective?

3. How are different homework purposes related to the amount ofhomework that is completed?

4. How are the class variables, index of parental education andindex of students engagement on homework, related to math-

ematics achievement?

12 P. Rosrio et al./Contemporary Educational Psychology 43 (2015) 1024
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3. Method

3.1. Participants

Our study focused on the 6th grade level. Sixth grade was se-lected because it is the final grade level of elementary school in thePortuguese educational system, and students complete a nationalstandardized exam in mathematics at the end of the school year(June), which counts for 30% of the students overall mathematicsgrade. Sixth-graders have three mathematics lessons per week for90 minutes each and typically spend a large portion of the schoolyear reviewing concepts and skills that were taught in elementaryschool in preparation for the national exam in June. Each year, el-ementary schools are ranked according to their students results.Thus, school administrators and teachers from the mathematics de-partment seek to understand practices that could enhance theirstudents results on the exam. School administrators and teachersalso want to learn how to best prepare their students for junior highschool, where students are expected to learn higher-level mathe-matics concepts.

Our sample consisted of 27 mathematics teachers from the 6thgrade, including 20 women and 7 men between 28 and 54 yearsold (M= 38.67; SD = 8.22). These mathematics teachers teaching ex-

perience ranged from 5 to 30 years (M= 19;SD = 8.81). Participantsincluded 638 6th graders who attended seven public schools innorthern Portugal. Of these students, 321 (50.3%) were girls and 317(49.7%) were boys, with ages that ranged from 10 to 13 years(M=11.41;SD =0.62). Finally, 211 students were in the first treat-ment condition (i.e., practice homework purpose), 208 in the secondcondition (i.e., preparation homework purpose), and 219 were in-cluded in the third condition (i.e., extension homework purpose).Three hundred and four (47.6%) parents of the participants had com-pleted the 9th grade (Portuguese compulsory education), 95 (14.9%)completed the 12th grade, 199 (31.2%) held a degree, and 40 (6.3%)did not respond.

3.2. Instruments and measures

3.2.1. Adjustment variables

3.2.1.1. Prior knowledge of mathematics. Previous knowledge (whichserved as a pre-test and a control variable) was obtained from stu-dents grades on a final mathematics exam that was conducted theprevious year. Data were collected in the schools secretariat. In Por-tuguese compulsory education, grades are 1 and 2 (negative), 3(passing), 4 (good), and 5 (excellent).

3.2.1.2. Mathematics self-efficacy. Self-efficacy in mathematics wasassessed with 10 items that were adapted fromJot, Usher, andBressoux (2011).FollowingBanduras (2006)guidelines for con-

structing self-efficacy scales, the current study phrased the 10 itemsto assess the students perceived skills in computation, numeracy,and geometry (e.g., I know arithmetic properties, for example, thecommutative property of addition). To adjust the items to thecontent that the students should have already known, the same teamof mathematics teachers that created the final non-standardized test(i.e., two faculty members from the Mathematics Department andthree 6th-grade school teachers) helped the researchers phrase theitems. The final list of items was evaluated by five experts, and thegeneral diagnostic reliability was excellent: =.93 (p 60 minutes. On the following Monday,the students delivered the 6 daily logs to the research teams in asealed envelope.

3.2.1.5. Parents educational level. In accordance with Xu (2015), stu-dents responded to two items about their parents education level

(one for the father or guardian and another for the mother or guard-ian). The students responded to one of the following options for bothitems: until 9th grade (Portuguese compulsory education),until 12th

grade,graduate, and postgraduate. A composite variable for paren-tal education was obtained by averaging these two items (Cronbachs =.78).

3.2.2. Outcome variables

3.2.2.1. The mathematics achievement testMAT(description). Thisinformation (which served as a post-test) was obtained based onthe students grades on a final non-standardized mathematics examthat was specifically designed for this study by two faculty membersfrom the Mathematics Department and three 6th-grade schoolteach-ers. The exam focused on geometry content that was reviewed in

the homework that was assigned during the research study. Thequestions on the exam were organized into two sections (see twoexamples for each section in theAppendix). The first section con-sisted of 5 multiple-choice questions that evaluated studentsknowledge about geometry (i.e., dots, lines, geometrical figures andsolids), and the second section had 5 mathematics problems thatwere about perimeters, areas, and the volumes of geometrical areasand solids (two of the problems involved questions that were relatedto daily life routines). The duration of this exam was 45 minutes.For the purpose of this study, each question was graded with a di-chotomy as follows: 1 represented the correct answer and 0represented the wrong answer. The final score was divided by 2 toadjust the result to Portuguese Compulsory Education guidelineswith grades that ranged from 1 to 5, as was explained above.

3.2.2.2. Psychometric properties of the MAT. Item response theory(IRT, which is also called latent trait theory) provides a valuablemethod for analyzing the psychometric properties of an outcomemeasure. As Lord (1980) notes, IRT begins with a mathematical state-ment as to how the responses to the questions on the test dependon the level of ability or skill of the examinees. In the current study,the type of grading (the binary response format, right/wrong) andthe model fit statistic helped to select a uni-dimensional two-parameter logistic (2PL) model. This model estimates the probabilityof correctly answering the item as a logistic function of the differ-ence between individual levels of knowledge, the slopes (or thediscrimination parameters) and the item difficulty (or the inter-cept parameters). According to the GCONV =0.00000001 criterion,

optimal convergence was obtained with 17 iterations using SAS PROC

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IRT 9.4 TS1M2 (SAS Institute, Inc, 2014). The deviance (2 timesthe log likelihood) for the 2PL model was 3385.37 while the devi-ance for the 3PL model was 3363.09. A likelihood-ratio test comparedthe two models and indicated that the 3PL model was signifi-cantly different from the 2PL model (2 =22.28, df=9). However,because the standard errors for several of the guessing parameterestimates were close to zero, we chose the 2PL IRT model.

Descriptive statistics, Cronbachs alpha reliability coefficients, theeigenvalues from the polychoric correlation matrix, and the esti-mated item parameters are presented in Table 1.Our goal was toassess students mathematics achievement, and, thus, we used a uni-dimensional IRT model. The eigenvalue for the first factor of thepolychoric correlation matrix was 5.92 times the second, which sug-gests that a one-factor model is sufficient. Moreover, the Scree Plotshowed that the first factor explained a greater percentage of thevariance (almost 52%). Cronbachs alpha for the test items was 0.78,and the standard error of measurement (SEM) for this reliability es-timation was 0.47.Table 1also shows the parameterizations for the10 items of the MAT according to the 2PL model. The slope param-eter values ranged from 0.9 to 2.3, which suggest that all items

adequately measured the latent trait, and the threshold param-eters ranged from 3.14 to 1.25.

Table 2presents the values for the chosen ability levels (: 4,3, 2, 1, 0, 1, 2, 3, 4) for the item response functions (IRF) or itemcharacteristic curve (ICC), the test characteristic function (TCF), theitem information functions (IIF), the test information function (TIF),and the standard error of measurement (SEM) for each level of thelatent trait. For dichotomous items, the IRF is the probability of acorrect response on the item. For example, the IRF indicates thatwhen is zero, which is average, the probability of correctly an-swering Item 10 is almost 0.11. When is 2, the probability is almostzero, and when is +2, the probability increases to 0.99. The IRTanalysis also provides information about the skill level at which anitem is more informative. The IIF indicates the precision to whicheach item measures the different trait levels, and, hence, providesan alternative technique for selecting items. Among the 10 items,two (item3 and item9) had slope values that were slightly less than1, which suggests that these two items provided less informationthan the other items. Finally, there are two important aspects of theTIF 2PL model. First, the MAT test produced its maximum

Table 1

Descriptive statistics and item parameter estimates.Item Mean SD Alpha Slope SE t Difficulty SE t

1 .873 .333 .771 2.145 .3275


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information at approximately zero on the latent-trait variable;however, it had an asymmetrical distribution for ability level, whichindicates that the scale works best for individuals with average orbelow average levels of mathematical competency. Second, themeasure is slightly less precise for above-average ability levels.

3.3. Procedure

First, we obtained consent from the Portugal Ministry of Edu-cation to conduct the present study in the schools. After obtainingconsent, the research team contacted 15 public schools in the north-ern part of Portugal. Seven schools that were located in an urbanschool district in the northern part of Portugal agreed to partici-pate in our research. In these schools, the families of the studentsare globally lower-middle class because of the high percentage ofstudents (ranging from 38.2% to 43% in the seven schools) whoreceive free or reduced-price lunches. These demographic data werecollected from the participating schools offices.

Next, all 35 6th-grade mathematics teachers from the enrolledschools were invited to participate in the research through an emailthat explained the overall study objectives. Twenty-seven teach-ers (a response rate of 77.1%) communicated their intention toparticipate via email. Then, a total of 702 students parents fromthe enrolled were informed about the study aims and proceduresthrough a letter and were asked for permission for their childs par-ticipation. Six hundred and thirty-eight students returned signedparental consent forms (a response rate of 90.8%), and only thosestudents were allowed to participate in the study. Participation wasvoluntary for teachers and students, and participants confidenti-ality and anonymity were assured (e.g., eliminating the names andresearchers personal notes that could link the participants to theirteachers or schools). Finally, the 27 teachers (classes) who agreedto participate (with their students who assented to participate) wererandomly assigned to three different homework purpose condi-tions (i.e., practice, preparation and extension), with 9 teachers ineach treatment group (for details, seeTable 3).

Two weeks prior to beginning the study, the volunteer teachersparticipated in a 4-hour informational meetingto discuss the projectgoals and the research design in detail (e.g., information on thenumber of homework assignments, types of homework purposesand tasks that were in accord with the three purposes, and gradedhomework as a follow-up practice). Threeweeks post-intervention,theteachers attended a two-hour evaluation meeting toanalyze theirexperiences whileparticipating (e.g.,comments and suggestionsthatcould help in future research) and discuss preliminary data. Teach-ers who fully participated in the research were offered a 27-hour (1ECTS) training course about learning and instruction processes thatwas conducted at the University (Universidade do Minho).

The data were collected in March and April. The three experi-mental conditions reflected the instructional homework purposesthat are defined in the literature (Cooper, 2001; Epstein & Van

Voorhis, 2001; Lee & Pruitt, 1979) as follows:

1) Practice homework (Epstein & Van Voorhis, 2001; Lee & Pruitt,1979) aims topractice skills taught inclass,increase speed,dem-onstrate mastery, retain skills, review work, and study for tests(Epstein & Van Voorhis, 2001, p. 182). In this condition, the en-rolled teachers assigned two types of homework assignments:Thestudents hadtoreview thematerial that was coveredin classand write five main ideas about it in their notebooks (home-work assignments 1 and 3); and in the other four homeworkassignments, the students had to complete exercises from theirtextbook while focusing on the content that had been coveredin class (homework assignments 2, 4, 5, and 6). For example, inhomework 4, at the end of the topic on geometry (i.e., geomet-rical volume), the students completed the following exercise:Calculate the volume of a cube whose side is 5 cm.

2) Preparation homework(Epstein & Van Voorhis, 2001; Lee &Pruitt, 1979)aims to prepare students for the next class. Thehomework can be designed to stimulate students thinkingabout a topic, for example, when teachers ask students tooutline ideas for an essay that will be written in class (Epstein& Van Voorhis, 2001, p. 182). In our study, there were twodifferent types of assigned homework. In three of the home-work assignments (i.e., 1, 3 and 5), students had to learn thetopic for the next lesson in their textbook and prepare a brief

oral presentation to present in class. In the other three home-work assignments (i.e., 2, 4 and 6), students had to study thetopic that would be discussed in the next lesson in their text-book and write five main ideas in their notebooks. Forexample, in homework 4, the students read about geometry(i.e., geometrical volume) in their textbook and wrote the def-inition of volume and the different types of geometrical solidsthat they learned in class in their notebooks.

3) Extension homework (Cooper, 2001) aims to promote thetransfer of learning to new tasks. According toLee and Pruitt(1979),extension assignments differ from practice assign-ments in the degree of application in the abstract thinkingrequired. Practice assignments require very little abstractthinking (p. 32). In this study, students had to solve twomath-

ematics problems in their notebooks for each of the sixhomework assignments. For example, in homework 4, the stu-dents had to find the answer to the following mathematicalproblem: The pot that Antnios family uses in the kitchenis 10 cm high and has a 15-cm radius. Calculate the approx-imate value of the pots volume and determine how manyliters of soup Antnios family can serve for dinner using thatpot. Show how you reached your answer.

Prior to starting the research, university teachers from the De-partment of Mathematics, the participating teachers and the researchteam analyzed whether the tasks that were to be assigned wererelated to the three purposes, asEpstein and Van Voorhis (2001)suggested, and worked to calibrate the level of difficulty of the tasks

that were assigned for each condition.

Table 3

Distribution of student and treatment by teacher and school.

School_1 School_2 School_3 School_4 School_5 School_6 School_7

Teacher1(T_HW2)njk =26



























Note:T_HW1 (practice homework), T_HW2 (preparation homework), T_HW3 (extension homework).



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In the Portuguese school system, 6th graders have three 90-minute mathematics lessons per week. In this study, teachers wereasked to only assign homework in the first class of the week. Overthe six weeks, all teachers taught the topic of geometry (i.e., points,lines, geometrical areas and volume), and each week, the home-work that was assigned focused on the same content despite theexperimental condition (i.e., practice, preparation, extension). In eachexperimental condition, the weekly homework that was assignedwas the same in all classes.

In the week prior to starting the intervention in the class, thequestionnaires and measures (i.e., mathematics self-efficacy, math-ematics study time and parental educational level) wereadministered to students during regular classes in the second term.For each homework assignment in the three conditions, the pro-tocol was as follows: at the end of the class, to clearly state thehomework assignment, students were asked to write the home-work assignment in their notebooks. In the next lesson, teachersbegan the class by checking whether the students had completedthe homework by completing a homework diary log. To control forthe influence of the homework follow-up practices on achieve-ment, all homework assignments were collected and graded by theteachers using the Portuguese grading system for elementary school(i.e., 1 and 2negative, 3passing, 4good, and 5excellent). During

the subsequent lesson, the graded homework was returned to thestudents. There was an exception for the preparation homework as-signments 1, 3 and 5 (i.e., students had to learn the topic for thenext lesson in their textbook and prepare a brief oral presentationto present in class). In these assignments, the teachers gave a grade(using the same grading scale) after the students oral presenta-tion of their homework in class. At the end of the sixth week, allstudents completed the same non-standardized mathematicsachievement test (see the measures section).

To ensure compliance in each experimental condition, teach-ers were required to send an email to the first author of the studyat the end of the second class of the week that reported whetherthe protocol had been followed. At the end of the study, each teachersent their homework diary log to the first author and received a

summary table for confirmation purposes. To further examine themeasures reliability (i.e., whether the teachers followed their home-work purpose condition protocol), five students from each class wererandomly chosen to receive an email in which they were asked toreply by attaching the homework task that was assigned by theteachers. Fleisss Kappa statistic(Fleiss, 1981)indicated that thegeneral diagnostic reliability among the randomly chosen stu-dents from each class was excellent: =.95 (p


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the intra-school correlation, which measured the proportion of thevariance between the schools, and the intra-teacher correlation,which measured the same between teachers. We also analyzedwhether the level-3 predictors varied across schools.

4. Results

4.1. Descriptive analyses

Pearson correlation coefficients, means and standard devia-tions for the variables that were included in the model are presentedinTable 4.Based on the variables averages, we conclude that self-efficacy and mathematical achievement levels (pre and post) areclose to the average. In addition, students completed most of theassigned homework. Moreover, students spent an average of 15minutes per day studying (excluding homework time). Finally, par-ents average educational level was less than 12th grade.

4.2. Multilevel analyses

The analyses were performed using a set of 2-level HLMs becausethere was not a sufficient number of schools to conduct a full 3-levelHLM analysis. However, empirical justification was provided prior

to conducting the 2-level HLM analysis. To examine the potentialeffects of the different schools, we fit a three-level unconditionalHLM model and a model with school-level predictors.

4.2.1. Three-level analysesAlthough the current study does not include any school-levelvari-

ables in the model, student-level data for PEL (i.e., parent educationlevel), MAT1 (i.e., prior mathematics achievement) and AHWC (i.e.,homework engagement) were aggregated to describe the set of 6th-grade students in each school. The results from the three-levelunconditional HLM and a model with school-level predictors arepresented inTable 5.

The estimated overall mean across schools was 3.37 and was sig-nificantly different from zero (p


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conduct a three-level analysis due to the small number of schools,provide an argument for using a simpler, two-level analysis withstudents at level 1 and teachers at level 2, ignoring the schoolseffects.

4.2.2. Two-level analysesTo address our first goal (i.e., to examine the impact of teach-

ers homework purposes on students mathematics achievement),we examined a fully unconditional model that separated the vari-ance from mathematics achievement into between and within-class components.Table 6shows the results from fitting the modelto the data. The estimated mean on mathematics achievement inthis sample of classes (3.3646) is different from zero (p < .001).However, it is notable that there were statistically significant dif-ferences in average student mathematics achievement levels betweenclasses (u0j =0.127;p


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evidence of a statistically significant change in students meanachievement scores due to the aggregate amount of homework com-pleted and the class-level aggregate parents education. In addition,there was no evidence that the amount of homework completeddiffered depending on the type of homework purpose.

As previously noted, a useful method for testing pairwise differ-ences between groups in a manner that is consistent with theresearchobjectives is to perform a series of contrasts.The BenjaminiHochbergFDR controlling procedure was adopted to assess the statistical signif-icance, andTable7shows the significant contrasts atp < .05.

Post hoc tests revealed that there was a significant difference

between the T_HW2 and T_HW3 groups (t= 3.10, p =.0146) anda marginally significant difference between the T_HW1 and T_HW3groups (t= 2.04,p =.0793) (seeTable 7andFig. 1). The differencebetween the T_HW1 and T_HW2 groups was not significant (t= 1.02,

p = .3180), which indicates that the treatment that was responsi-ble for a change from pre- to post-test is T_HW3 (extensionhomework).

To analyze the interaction between the type of homework pur-poses (T_HW) and the amount of homework completed (AHWC)and its effect on mathematics achievement (our third goal), we di-chotomized the AHWC (lowwhen AHWC 4 andhighwhen AHWC>4). Table 8 shows the descriptive data forthis analysis and that therewas no interaction between the amount of homework assign-ments completed and the type of homework purpose. As previously

noted, the type of homework purpose differentially relates to math-ematics achievement.

4.3. Ancillary analysis

Despite difficulties in controlling for variables in the school set-tings, there is a general call for studies that use experimental designsand are conducted in holistic educational environments (e.g.,Dettmers et al., 2011; Xu & Wu, 2013). Although there were diffi-culties conducting the current study (see section 6), our preliminary

results are promising and provide important educational insightsthat justify additional analyses. To further our understanding ofhomework purposes and the study dynamics, we organized a post-research evaluation meeting to analyze the participating teachersexperiences. Consistent with McInerney (2012), we conducted a qual-itative analysis of notes that were completed by the authors of thisresearch about the participants interventions. The discussion of thesefindings will be limited; however, we believe that the topics dis-cussed below will help calibrate our interpretations and deepen ourunderstanding of the complex process of homework purposes.

4.3.1. The design

It was difficult to recruit schools to participate in the study. Sixth-grade teachers are often intensely focused on preparing their

students for the national final exam, and thus, they had a limiteddisposition toward embracing research projects that would consume

Table 7

Post hoc contrasts using the BenjaminiHochberg procedure for FDR control to iden-tify the best types of homework purposes.

Groupcomparison

Estimate S E DF tValue p Fdr_p Partial 2

T_HW2-3 .1891 .0609 24 3.10 .0049 .01458 .270T_HW1-3 .1265 .0621 24 2.04 .0529 .07933 .181T_HW1-2 .0626 .0614 24 1.02 .3180 .31799 .037

Note: T_HW1 (practice homework), T_HW2 (preparation homework), T_HW3 (ex-

tension homework).

Fig. 1. Graphical representation of changes in the mean of academic achievement in mathematics from pretest to posttest for each type of homework purpose.

Note: T_HW1 (practice homework), T_HW2 (preparation homework), T_HW3 (extension homework).

Table 8

Cell means and standard deviations on the mathematics achievement test for threehomework purposes as a function of the amount of homework completed.

T_HW AHWC Final mathematicsachievement

n M DS

Practice homework Medium-Low 49 2.92 0.86Medium-High 161 3.40 0.88

Preparation homework Medium-Low 27 2.96 0.98

Medium-High 181 3.35 0.90Extension homework Medium-Low 7 3.00 1.15

Medium-High 212 3.54 1.01

Note: T_HW (type of homework purposes); AHWC (Amount of homework assign-ments completed).

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time and effort. Initially, the participating teachers were notcompletely comfortable with the proposed random selection cri-teria. None had previously participated in a controlled school-based investigation,and thus, it was important to discuss the featuresof an experimental design and the associated gains for their prac-tice to increase adherence with the design. The following statementillustrates teachers opinions toward participating in the currentstudy: Im thankful for participating in this research because overmy fourteen years of teaching experience I never thought about thepurposes of my homework like this. I realize now, I had a more in-tuitive approach (T14).

The collaborative work that was completed in the informa-tional meeting for the types of homework tasks assigned in relationto the intended purpose was also an important aspect that was ref-erenced by participants: Calibrating homework tasks and aligningtasks with the purposes was a very good experience. . .. For my stu-dents I never check on this relationship For sure, it is possible tofind several mismatches in my daily practice. . .. I clearly need tobe attentive, this research alerted me to the importance of this topic(T9). Several teachers also highlighted the importance and meritsof assigning homework with different purposes than those that theyusually assign. For example, one teacher said: I was not willing toparticipate in this research, but my friend convinced me. . .. Besides,

my experimental condition [extension homework] was not mynatural condition, do you understand what I mean? I do not assignextension homework often, and I was not happy when I found thatI was randomly assigned to that condition. After a brief reflectionI decided to go with the flow, but soon I realized the merits of as-signing extension homework Students were more engaged, Ibecame more engaged. . . (T4).

4.3.2. Homework purposesIn the post-research evaluation meeting, participating teachers

had the opportunity to reflect on their perspectives of the purposeof homework. For example, some teachers believed that it was im-portant to adjust the type of assigned homework to address theirstudents educational needs. However, daily instructional con-

straints (e.g., the high number of students per class, the limited timeavailable for homework follow-up, the pressure for preparing stu-dents for the final exam and to review all of the content) made itdifficult for teachers to prepare and design homework tasks thatmatched students learning needs.

In line with the literature (e.g.,Danielson et al., 2011), most ofour participants reported that they mainly used a single type ofhomework: I always assign homework to instigate students to prac-tice the contents worked in class (T9). Several teachers explainedreasons for this routine, as follows: I believe that assigning home-work according to my students characteristics would be beneficialfor them. In fact, I would like to assign tailored homework address-ing students struggling with learning specific needs but also highachievers needs; often we forget their needs. . .. However, Im teach-

ing six classes, four from the 5th and two from the 6th grade, andthat makes it impossible. I would need extra time to prepare home-work and then to grade it for the following class, time that I definitelydont have. So, I end up assigning homework to help them prac-tice the basic contents covered in class. Every student will be ableto do it and for me it is easier. If I assign problem solving tasks, themajority of students will struggle to solve the exercises, and in thenext lesson Ill need a lot of time to explain it in class, time that Ineed to cover the content, you see?! (T26).

5. Discussion

This study investigated the relationship between three home-work instructional purposes and mathematics academic

achievement. To improve the reliability for the analysis, we

controlled for the effects of six homework-related variables (i.e.,gender, prior achievement, mathematics self-efficacy, amount ofhomework, amount of time studying mathematics, and parental edu-cational level) that the literature indicated are important toexplaining a significant amount of variance in mathematics per-formance(Fernndez-Alonso et al., 2015; Hemmings et al., 2011;Kitsantas et al., 2011; Lee et al., 2014; Rosrio et al., 2009; Trautwein,Niggli et al., 2009; Zuffian et al., 2012) (seeTable 6). Data from therandomized pretestposttest clustered design were analyzed withmultilevel regression models that assessed whether different typesof homework purposes impacted students mathematics achieve-ment, which type of homework is most effective, how thisrelationship could be modulated by the amount of homework com-pleted, and how the two Level-2 aggregated parental education andhomework completion covariates were related to mathematicsachievement.

The unconditional means model suggested that teachers dif-fered in their students average mathematics achievement scoresand that there was even more variation among the students withinteachers. In fact, the variance component within teachers was nearlysix times the size of the variance component between teachers. Theconditional 2-level HLM analysis revealed that the variance in thestudent level residual errors was almost three times smaller than

the original when the model included prior achievement, mathe-matics self-efficacy, the amount of homework, and parentaleducational level as explanatory variables. In contrast, the resultsalso revealed that the residual variance between teachers was sub-stantially smaller than the original when the explanatory variablethat represented the type of homework purposes was added to themodel.

For our first goal, after the six sessions, the homework datashowed statistically significant differences in mathematics achieve-ment when comparing homework purposes (i.e., homeworkextension vs preparation homework or practice homework) in favorof homework extension. According to our findings, homework thathas the purpose of extension (e.g., promoting problem solving skills)is beneficial for improving students achievement in mathematics.

Our data complement that ofZhu and Leung (2012).AnalyzingTIMSS 2003 data, these authors(Zhu & Leung, 2012) found thathomework that was focused on problem solving had a significantpositive effect on students mathematics achievement. The othertypes of homework that were analyzed in that study (i.e., gather-ing statistics data and reporting and finding one or more applicationsof the content covered) had no effects on students academic achieve-ment. However,Zhu and Leung (2012)identified that a limitationin their study was that the last two types of homework were lessprescribed than problem-solving (one type was not prescribed) andthat the TIMSS test questions were primarily focused on problemsolving skills. Thus, students were expected to have higher scoreson the TIMSS tests when they completed problem solving focusedhomework. To overcome this limitation from prior research, our

study designed the final exam (used as a post-test) that reviewedall of the skills that were enhanced by the prescribed homeworkassignments (e.g., the practice skills taught in class, transferring learn-ing to new situations). Furthermore, the PISA 2012 results (OECD,2014b) corroborated Zhu and Leungs (2012) findings, which showedthat problem-solving was associated with higher levels of schoolachievement in both mathematics (r=0.81) and science (r=0.78).Moreover, students who were open to solving mathematics prob-lems, that is, students who felt that they could handle the informationprovided while seeking explanations for the presented data andsolving complex problems (see OECD, 2014b, p. 18), scored 31 pointshigher in mathematics, on average, than those who were less opento problem solving.

Homework with the purpose of extension is often perceived by

students as academically demanding and research has found a



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positive predictive effect for this homework characteristic on aca-demic achievement at the class level (e.g., Dettmers et al., 2010) andat both levels(Trautwein et al., 2002).

These findings emphasize the importance of a homework prep-aration phase in which teachers calibrate the homework tasksdifficulty to match the students proximal zone of development(Vygotsky, 1978) and promote homework engagement (e.g., ad-

justing the task difficulty and matching the students educationalneeds). In fact, Zakharov, Carnoy, and Loyalka (2014) concluded thattasks that are adjusted for difficulty have a positive impact on stu-dents performance. Our participants reported that this practice wasvalued; however, in everyday classes, they face several constraintsin performing this instructional practice (e.g., the vast number oftopics to discuss in class during the school year, the number of stu-dents per class (an average of 26), and the number of classes theyteach at different grade levels (i.e., 5th, 6th and 7th grade). Thus,teachers often assign the same homework to all students most ofthe time.

In summary, the data indicated that when comparing the threehomework instructional purposes, students who completed home-work that had a purpose of extension had higher grades inmathematics. These results are aligned withTrautwein, Niggli et al.(2009),who found that homework that had a purpose of drilling

and practicing had a negative relationship with school achieve-ment. Moreover, the literature suggests that teachers most oftenprescribe homework that has a purpose of preparation and prac-tice (see Bang, 2012; Danielson et al., 2011; Kaur, 2011; Muhlenbrucket al., 2000). Our preliminary results should be further examinedin future research, even though the data indicate potentially inter-esting educational questions. Because higher grades in mathematicsare negatively related to more prescribed homework assign-ments, this issue is important for teachers, school administratorsand researchers to address. For example, a statement from one par-ticipating teacher, which was corroborated by several otherparticipants in the post-research evaluation meeting, may help un-derstand teachers reasons for assigning homework that had apurpose of practicing tasks: I know problem-solving is the home-

work task with more impact on achievement, but they [students]often struggle to complete those tasks, and I have to spend moretime to correct it in class, and I dont have that time. So I usuallyassign practice tasks (T3). Future studies should conduct in-depthanalyses on teachers reasons for assigning particular homeworktasks. Still, we believe that ourpreliminary findings could help teach-ers and school administrators to design homework policies andschool-based interventions for homework. For the second re-search aim, this study adds to the literature by comparing thepurposes of homework (the three that were analyzed) and their as-sociation with students mathematics achievement by examininghow completing more or less homework could affect the above re-lationship. The findings showed that despite the homework purpose(i.e.,practice, preparation or extension), it is always better to do more,

compared to less, homework. In summary, data from both goals in-dicated that the amount of homework did not affect the relationshipbetween homework purposes and academic achievement. Thus,when possible, mathematics teachers should prescribe home-work that has a purpose of extension.

Finally, we emphasize that the results were not significant whenexplaining mathematics achievement at the class level (i.e., whenconsidering Grp_PEL [parental educational level] and Grp-AHWC [theindex of students engagement in their homework]). In fact, it was ex-pected that the Grp-AHWC variable would be associated with asignificant percentage of the variance for students achievement ineach class, for example, as a result of the type of work in class orthe supervision and monitoring of that work (Rosrio et al., 2014a).This finding is educationally important because it indicates that there

is no relationship between the instructional processes in each of

the classes enrolled (e.g., monitoring the work performed in class)and the amount of homework that was completed by the stu-dents. Thus, completing more homework was primarily related tostudent-level personal characteristics (e.g., motivation, attitude andinterest,Bembenutty & White, 2013; Fernndez-Alonso et al., 2015;Xu, 2005), homework self-regulation (Hong, Peng, & Rowell, 2009),and homework management, including time management (Nezet al., 2013; Xu, 2010c), and less influenced by the teacher or theinstructional condition.

The last finding is aligned with several studies that found a smallrelationship between instructional behaviors and completing as-signed tasks (e.g.,Fernndez-Alonso et al., 2015; Rosrio et al., 2014a;Zhu & Leung, 2012). Additionally, with data-focused approaches forteaching (e.g.,Prosser & Trigwell, 2006; Rosrio et al., 2014b )thatused a multilevel design, teachers reported that their teaching ap-proaches influenced students approaches to learning, but theirinfluence was low(Rosrio et al., 2013).

6. Study limitations and future research

The data from this study should be examined while account-ing for several limitations.Our findings show that using a randomizedpretestposttest clustered design and multilevel modeling, one type

of homework purpose (extension) was more related to mathemat-ics achievement than the other two purposes (practice andpreparation). Although our data were analyzed at two levels (theindividual and class levels), it should be noted that the number ofunits on the second level (class level) was limited and may havesignificantly impacted the estimated parameters and their associ-ated probabilities (seeVallejo, Fernndez, Livacic-Rojas, & Cuesta,2015; Vallejo, Tuero-Herrero, Nez, & Rosrio, 2014). For example,if the number of classes had been greater than 30, it is possible thatthe T_HW1-3 comparison (practice versus extension homeworkpurpose) would have reached statistical significance atp < .001 (seeTable 6).

Moreover, this study examined only mathematics achievementas an outcome variable. It would be informative and useful for future

research to explore the relationship between homework purposesand the other variables that are related to students homework be-haviors (e.g., homework procrastination, homework effort, time spenton homework, homework completion rate, and homework feed-back) and students homework performance (e.g., quality of thehomework presented in class). Moreover, it could be interesting toexamine whether students perceptions of the homework purposeinfluence their actual homework behaviors. For example,Rosrioet al.s (2013)study on approaches to teaching concluded that stu-dents perspectives of their teachers approaches to teachinginfluenced their own approaches to learning and studying. It is pos-sible that several of the homework purposes may have been bettersuited for some students than others, depending on their previ-ous achievement levels and academic goals (Valle et al., 2013). Thus,

future studies should also examine the degree of individualiza-tion in students homework while accounting for students priorknowledge and other motivational aspects (e.g., self-set goals). Inthe current study, the number of homework assignments com-pleted by students and the homework follow-up practices that wereused by all of the teachers (i.e., return the homework graded in thefollowing class) were controlled for to reduce their possible influ-ence on achievement (i.e., teachers sent an e-mail reporting whetherthe protocol had been followed and five students in each class re-ceived an e-mail with the same purpose). However, we cannotguarantee that the teachers did not use other homework follow-up practices in class (e.g., orally correcting homework), which wouldcompletely discard the effect of these non-controlled homeworkfollow-up practices on the results. Future studies should address

this limitation by controlling for the effect of teachers homework



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follow-up practices in class(Rosrio et al., 2014a). Moreover, thisstudy was conducted with students from the 6th grade; thus, it isimportant to study homework purposes at other school levels andover a period of time to learn and compare the results with thecurrent study. Finally, two measures in our study relied on self-report data (i.e., amount of time studying mathematics andmathematics self-efficacy), which allowed for collecting informa-tion about students perceptions of the learning processes but didnot capture real-time responses in authentic learning environ-ments or multiple reporters views of the same constructs (e.g.,parents reports of students study time). Future research shouldaddress these limitations by, for example, using on-task measuresand multiple sources that provide the same information. Futurestudies should further examine teachers perceptions of the home-work assignment design (e.g., purpose, frequency, and amount) andthe reasons that underlie their educational choices. The results ofthis further research could help to build an in-depth understand-ing of teachers perspectives during the first step of the homeworkprocess (i.e., planning and design) and the possible constraints onteachers practices with respect to homework.

Finally, our study only focused on the teachers role in the firststep of the homework process (i.e., homework preparation anddesign). Our findings show that homework purpose is an impor-

tant homework variable, but it does not explain the complexrelationship between doing homework and academic achieve-ment. Future research could explore other homework characteristics(e.g., the homework quality, seeDettmers et al., 2010), in additionto other homework behaviors that are the teachers responsibility,such as grading homework and giving feedback (e.g.,Cooper, 2001;Nez, Surez et al., 2014), as they may influence the relationshipbetween homework purposes and academic achievement.

7. Conclusions and implications for practice

Our data indicated that homework for the purpose of exten-sion was the most effective type of homework. This finding iseducationally important because homework assignments that have

the purposes of drill and practice are most often prescribed by teach-ers (e.g., Danielson et al., 2011). In the post-research meeting,participating teachers defended the extension homework assign-ments as being the most powerful for learning; however, three factors(i.e., the limited amount of time for monitoring students home-work, the large number of students in each class, and studentsdifficulties with these types of exercises) were important justifi-cations for not assigning this type of homework more often.

Thus, teachers should acknowledge that homework with thepurpose of extension is complex. To promote the efficacy of this typeof homework purpose, teachers should provide specific instruc-tions, solid guidelines and an appropriate amount of time forcompletion to not burden students with extra academic tasks or un-realistic deadlines (Lee & Pruitt, 1979). The following statement from

one participant indicates the need to help teachers reflect on theirpractices, which in this case was homework related: This was thefirst time I looked at homework purposes with close attention. I hadnever stopped to carefully reflect on my assignments and their goals(T7). School teachers with the support of school administrators andschool psychologists should consider analyzing the types of home-work that are usually assigned and reflect on their impact onstudents learning and achievement. To empower teachers in thehomework process, it could be useful to conduct school-based train-ing that addresses different aspects of the homework design (e.g.,the homework purpose, types of tasks to match each purpose, andanticipation of the time needed to complete the tasks for each typeof homework purpose). Our findings should be interpreted and ad-

justed for the context of each class. It is important for teachers to

build a deep understanding of their students educational status (e.g.,

prior knowledge gaps) and design homework that fits their stu-dents learning needs (e.g.,Cooper, 2001).

Furthermore, teachers could help students define their ownhomework goals, by emphasizing the importance of homework tasksin helping students achieve their learning goals (Nez et al., 2013).These educational initiatives could provide essential information fordesigning more effective homework practices to improve, forexample, students homework engagement and consequently theiracademic achievement.

Acknowledgments

This study was conducted at Psychology Research Centre, Uni-versity of Minho, and supported by the Portuguese Foundation forScience and Technology and the Portuguese Ministry of Educationand Science through national funds and when applicable co-financed by FEDER under the PT2020 Partnership Agreement(UID/PSI/01662/2013).

Appendix

Examples of the questions from the non-standardized Mathe-matics Achievement Test.

Structure Type ofquestion

Example

Firstsection

Multiplechoice

Joana drew an acute-angle triangle.Select one of the following amplitudes oftriangle angles that Joana drew.a) 30, 90, 60b) 30, 95, 55c) 35, 85, 60d) 35, 110, 35

Secondsection

Mathematicsproblems

The following figure shows a can of paint, in theshape of a cylinder, with a 30-cm diameter baseand a height of 28 cm.[figure]Calculate the capacity in liters of the can ofpaint.Present the results rounded to the nearest unit.Explain how you reached your answer.(Use =3.14)(Note: 1 liter =1 dm3)

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