definition of homework in mathematics

Homework Help
Article Directory
Math homework: What to expect and why it is important
/

Math Homework: What to Expect and Why IT Is Important

Parents across the country are starting to question the impact that math homework has on their children. This article discusses why math homework is important and what parents should expect to see in their children's assignments.

School boards around the country are beginning to place homework restrictions on K-12 grade public school teachers. These restrictions are being created in response to increasing instances of homework anxiety in students across the nation. However, before you approach your school board about creating similar restrictions, it is important to learn and understand the value of homework.

Math Homework

Math homework is any task assigned to students to complete outside of their math class, and is created to help students prepare to learn new mathematical concepts, practice ones that have already been introduced, and explore other math skills. These out-of-class assignments are help to reinforce the lessons a child is introduced during the school day.

Studies are finding a relationship between homework and student achievement in school. There has not been established a cause and effect relationship, but there is a strong correlation between the two traits. However, it is also acknowledged that these assignments are only effective when the math teacher takes time to prepare quality assignments that relate to the specific skills that students should be learning or practicing.

Too Much Math Homework?

Statements released by the National PTA and the National Education Association, kindergarten to third grade students should be assigned no more than 20 minutes total homework per day. Fourth to sixth graders should be assigned 20 to 40 minutes while seventh to twelfth grade teens have a varying recommended amount of homework per day depending on the difficulty of their courses. These figures are for all subjects combined, not just the mathematics course.

If your child is taking much longer than these recommended amounts of time on their homework, it could be for two reasons. Either your student's teachers are assigning too much homework or your child does not fully understand their assignments. If you are concerned that a teacher is overloading your child with homework then you should schedule an appointment to discuss this problem. If your child is struggling with their work due to a math skills gap then the answer may be outside tutoring. Busy families can help a child catch up with their peers through online tutoring. Tutoring centers that are located on the World Wide Web are professional and use the same proven methods as their conventional counterparts.

We Found 7 Tutors You Might Be Interested In

Huntington learning.

What Huntington Learning offers:
Online and in-center tutoring
One on one tutoring
Every Huntington tutor is certified and trained extensively on the most effective teaching methods
What K12 offers:
Online tutoring
Has a strong and effective partnership with public and private schools
AdvancED-accredited corporation meeting the highest standards of educational management

Kaplan Kids

What Kaplan Kids offers:
Customized learning plans
Real-Time Progress Reports track your child's progress
What Kumon offers:
In-center tutoring
Individualized programs for your child
Helps your child develop the skills and study habits needed to improve their academic performance

Sylvan Learning

What Sylvan Learning offers:
Sylvan tutors are certified teachers who provide personalized instruction
Regular assessment and progress reports

Tutor Doctor

What Tutor Doctor offers:
In-Home tutoring
One on one attention by the tutor
Develops personlized programs by working with your child's existing homework
What TutorVista offers:
Student works one-on-one with a professional tutor
Using the virtual whiteboard workspace to share problems, solutions and explanations

Find the Perfect Tutor

Our commitment to you, free help from teachers, free learning materials, helping disadvantaged youth, learning tools.

Make learning fun with these online games!
Looking for ways to bring learning home? Check out our blog.

Want to Help Your Child Learn?

Towards a theory of mathematics homework as a social practice

Published: 26 May 2013
Volume 84 , pages 371–391, ( 2013 )

Cite this article

Mara G. Landers 1

1952 Accesses

11 Citations

1 Altmetric

Explore all metrics

This article presents a theoretical conceptualization of mathematics homework as a social practice. Rather than considering homework as a task or an artifact, this approach frames homework in terms of the social contexts in which students participate and how students participate in those contexts. This perspective has long been suggested by homework researchers but has not been developed as a framework for understanding homework. Drawing from Wenger’s ( 1998 ) social theory of learning and research grounded in sociocultural theory, this conceptualization makes central meaning making and identity development, and puts forth meaning and identity as lenses for understanding students’ participation in the practice of mathematics homework. To further develop this conceptualization of homework, I draw on data from an ethnographic study of the role and meaning of mathematics homework in the lives of middle school students. Case studies of two students are presented to demonstrate the relationships among the meaning of homework, students’ identities, and their participation in the practice of homework. One student’s experiences support him in identifying as a capable mathematics student who is bound for continued academic success through high school to college. Thus, he comes to take on mathematics homework as a means to learn and succeed. The other student’s experiences support him in building an identity that leads him to reject homework. This conceptualization of homework and the case study data have implications for the practice of homework and for theories of students’ motivational dispositions in the context of mathematics homework and learning in general.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Movement in the Mathematics Classroom

Instruction and Construction of Mathematics at Home: An Exploratory Study

There exists a large and contentious body of research on the relationship between homework and learning and/or achievement. Many studies find that there is little or no value for younger students, but there may be some benefit for older students, especially under certain conditions, such as when the process involves teacher feedback. At the same time, achievement-focused homework research has been critiqued with respect to methodology (Trautwein & Köller, 2003 ) and the lack of consideration of homework practices (Landers, 2007 ). What is relevant to the current discussion, then, is that despite the ambivalence in research, homework is still a valued school practice for its potential or perceived benefits.

Lave’s work and the other research/theories discussed are grounded in anthropological and sociological concepts, which define identity and ways of interpreting human behavior very differently than traditional theories of motivation, which are grounded in individual/psychological concepts. See Nolen, Ward and Horn ( 2011 ) for a discussion of the problems and potential of bringing together these different theoretical traditions.

Roosevelt and the names of students and teachers are pseudonyms.

During the interview, I assumed that he meant “under the table.” This phrase makes sense in context.

Bempechat, J. (2004). The motivational benefits of homework: A social–cognitive perspective. Theory Into Practice, 43 (3), 189–196.

Article Google Scholar

Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning . Westport: Ablex.

Google Scholar

Bryan, T., & Nelson, C. (1994). Doing homework: Perspectives of elementary and junior high school students. Journal of Learning Disabilities, 27 (8), 488–499.

Chandler, J. (1983). Parents as teachers: Observations of low-income parents and children in a homework-like task . Cambridge: Harvard Graduate School of Education.

Chen, C., & Stevenson, H. W. (1989). Homework: A cross-cultural examination. Child Development, 60 , 551–561.

Clark, R. M. (1993). Homework-focused parenting practices that positively affect student achievement. In N. F. Chavkin (Ed.), Families and schools in a pluralistic society . New York: State University of New York Press.

Cole, M. (1996). Cultural psychology: A once and future discipline . Cambridge: Harvard University Press.

Cooper, H. (1989). Homework . New York: Longman.

Book Google Scholar

Cooper, H., Lindsey, J. J., Nye, B., & Greathouse, S. (1998). Relationships among attitudes about homework, amount of homework assigned and completed, and student achievement. Journal of Educational Psychology, 90 (1), 70–83.

Corno, L. (2000). Looking at homework differently. The Elementary School Journal, 100 (3), 529–548.

Corno, L., & Xu, J. (2004). Homework as the job of childhood. Theory Into Practice, 43 (3), 227–233.

Coutts, P. (2004). Meanings of homework and implications for practice. Theory Into Practice, 43 (3), 182–188.

Covington, M. V. (2000). Goal theory, motivation, and school achievement: An integrative review. Annual Review of Psychology, 51 , 171–200.

Dauber, S. L., & Epstein, J. L. (1993). Parents’ attitudes and practices of involvement in inner-city elementary and middle schools. In N. F. Chavkin (Ed.), Families and schools in a pluralistic society . New York: State University of New York Press.

Delgado-Gaitan, C. (1992). School matters in the Mexican-American home: Socializing children to education. American Educational Research Journal, 29 (3), 495–513.

Deslandes, R., & Rousseau, M. (2008). Evolution and relation of students’ homework management strategies and their parents’ help in homework during the transition to high school. Paper presented at the Annual Meeting of the American Educational Research Association, New York, March 2009.

Eckert, P. (1990). Adolescent social categories: Information and science learning. In M. Gardner, J. G. Greeno, F. Reif, & A. H. Schoenfeld (Eds.), Toward a scientific practice of science education . Hillsdale: Lawrence Erlbaum.

Eisenhardt, K. M. (1989). Building theories from case study research. Academy of Management Review, 14 (4), 532–550.

Else-quest, N., Hyde, J., & Hejmadi, A. (2008). Mother and child emotions during mathematics homework. Mathematical Thinking and Learning, 10 , 5–35.

Esmonde, I. (2009). Ideas and identities: Supporting equity in cooperative mathematics learning. Review of Educational Research, 79 (2), 108–1043.

Glazer, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research . New York: Aldine de Gruyter.

Hickey, D. T. (2003). Engaged participation vs. marginal non-participation: A stridently sociocultural model of achievement motivation. The Elementary School Journal, 103 (4), 401–429.

Hinchey, P. (1996). Why kids say they don’t do homework. The Clearing House, 69 (4), 242.

Holland, D., Lachicotte, W., Skinner, D., & Cain, C. (1998). Identity and agency in cultural worlds . Cambridge: Harvard University Press.

Hong, E., & Milgrim, R. A. (2000). Homework: Motivation and learning preferences . Westport, CT: Bergin & Garvey.

Horn, I. S. (2008). Turnaround students in high school mathematics: Constructing identities of competence through mathematical worlds. Mathematical Thinking and Learning, 10 , 201–239.

Howson, G. (2005). “Meaning” and school mathematics. In J. Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Mathematics education library: Vol. 37. Meaning in mathematics education (pp. 17–38). New York: Springer.

Chapter Google Scholar

Kemmis, S., & Grootenboer, P. (2008). Situating praxis in practice: Practice architectures and the cultural, social, and material conditions for practice. In S. Kemmis & T. J. Smith (Eds.), Enabling praxis: Challenges for education . Rotterdam: Sense Publishers.

Kralovec, E., & Buell, J. (2000). The end of homework . Boston: Beacon.

Landers, M. (2007). The homework cycle: Identity development and the meaning of mathematics homework in the lives of urban middle school students. Unpublished doctoral dissertation. University of California, Berkeley.

Lange, T., & Meaney, T. (2011). I actually started to scream: Emotional and mathematical trauma from doing school mathematics homework. Educational Studies in Mathematics, 77 , 35–51.

Larson, R., & Csikszentmihalyi, M. (1983). The experience sampling method. New Directions for Methodology of Social & Behavioral Science., 15 , 41–56.

Lave, J. (1997). The culture of acquisition and the practice of understanding. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives . Mahwah: Lawrence Erlbaum.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation . New York: Cambridge University Press.

Leone, C. M., & Richards, M. H. (1989). Classwork and homework in early adolescence: The ecology of achievement. Journal of Youth and Adolescence, 18 , 531–549.

Lerman, S. (2000). The social turn in mathematics education. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning . Westport: Greenwood Publishing Group.

Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46 (1–3), 87–113.

Martin, D. B. (2000). Mathematics success and failure among African-American youth: The roles of sociohistorical context, community forces, school influence, and individual agency . Mahwah: Lawrence Erlbaum Associates.

Martin, D. B. (2006). Mathematics learning and participation as racialized forms of experience: African American parents speak on the struggle for mathematics literacy. Mathematical Thinking and Learning, 8 (3), 197–229.

McDermott, R. P., Goldman, S. V., & Varenne, H. (1984). When school goes home. Teachers College Record, 85 (3), 391–407.

Nasir, N. (2002). Identity, goals, and learning: Mathematics in cultural practice. Mathematical Thinking and Learning, 4 (2&3), 213–247.

National Council of Teachers of Mathematics. (1980). An agenda for action . Reston: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics . Reston, VA: NCTM.

Nolen, S. B., Ward, C. J., & Horn, I. S. (2011). Motivation, engagement, and identity: Opening a conversation. In D. M. McInerney, R. A. Walker, & G. A. D. Liem (Eds.), Sociocultural theories of learning and motivation . Charlotte: Information Age Publishing.

Patall, E. A., Cooper, H., & Robinson, J. C. (2008). Parent involvement in homework: A research synthesis. Review of Educational Research, 78 , 1039–1101.

Pope, D. C. (2001). Doing school: How we are creating a generation of stressed out, materialistic, and miseducated students . New Haven: Yale University Press.

Pratt, M. W., Green, D., Macvicar, J., & Bountrogianni, M. (1992). The mathematical parent: Parental scaffolding, parental style, and learning outcomes in long division mathematics homework. Journal of Applied Developmental Psychology, 13 , 17–34.

Robinson, V. M., & Kuin, L. M. (1999). The explanation of practice: Why Chinese students copy assignments. Qualitative Studies in Education, 12 (2), 193–210.

Rogoff, B. (2003). The cultural nature of human development . New York: Oxford University Press.

Sfard, A., & Prusak, A. (2005). Telling Identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34 (14), 14–22.

Shumow, L., & Miller, J. D. (1991). Parents’ at-home and at-school academic involvement with young adolescents. Journal Of Early Adolescence, 21 (1), 68–91.

Shumow, L., Schmidt, J., & Kackar, H. (2008). Adolescents’ experience doing homework: Associations among context, quality of experience, and outcomes. The School Community Journal, 18 (2), 9–27.

Stake, R. E. (1995). The art of case study research . Thousand Oaks: Sage Publications.

Strauss, A. (1987). Qualitative analysis for social scientists . New York: Cambridge University Press.

Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory . Thousand Oaks: Sage Publications.

Street, B. V., Baker, D., & Tomlin, A. (2008). Navigating numeracies: Home/school numeracy practices . London: Springer.

Trautwein, U., & Köller, O. (2003). The relationship between homework and achievement—Still much of a mystery. Educational Psychology Review, 15 (2), 115–145.

Varenne, H., & McDermott, R. P. (1999). The Farrells and the Kinneys at home: Literacies in action. In H. Varenne & R. McDermott (Eds.), Successful failure: The school America builds . Boulder: Westview Press.

Vollstedt, M. (2011). The impact of context and culture on the construction of personal meaning. Paper presented at the Seventh Congress of the European Society for Research in Mathematics Education, Rzeszow, Poland.

Warton, P. (2001). The forgotten voices in homework: Views of students. Educational Psychologist, 36 (3), 155–165.

Wenger, E. (1998). Communities of practice: Learning, meaning, and identity . Cambridge: Cambridge University Press.

Xu, J. (2005). Purposes for doing homework reported by middle and high school students. The Journal of Educational Research, 99 (1), 46–55.

Xu, J. (2007). Middle-school homework management: More than just gender and family involvement. Educational Psychology, 27 (2), 173–189.

Xu, J., & Corno, L. (1998). Case studies of families doing third grade homework. Teachers College Record, 100 , 402–436.

Xu, J., & Corno, L. (2003). Family help and homework management reported by middle school students. The Elementary School Journal, 103 (5), 503–536.

Download references

Acknowledgments

This material is based upon work supported by the National Science Foundation under grant no. 0119732. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The author would like to thank Alan Schoenfeld, Maryl Gearhart, Mary Foote, Ilana Horn, and David Stinson for feedback on earlier versions of the article, as well as ESM’s editors and the anonymous reviewers for their thoughtful input.

Author information

Authors and affiliations.

Los Medanos College, 2700 E. Leland Rd, Pittsburg, CA, 94565, USA

Mara G. Landers

You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mara G. Landers .

Rights and permissions

Reprints and permissions

About this article

Landers, M.G. Towards a theory of mathematics homework as a social practice. Educ Stud Math 84 , 371–391 (2013). https://doi.org/10.1007/s10649-013-9487-1

Download citation

Published : 26 May 2013

Issue Date : November 2013

DOI : https://doi.org/10.1007/s10649-013-9487-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Sociocultural perspectives
Meaning making
Find a journal
Publish with us
Track your research

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Publications
Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

Advanced Search
Journal List
Front Psychol

“Homework Should Be…but We Do Not Live in an Ideal World”: Mathematics Teachers’ Perspectives on Quality Homework and on Homework Assigned in Elementary and Middle Schools

Pedro rosário.

1 Departamento de Psicologia Aplicada, Escola de Psicologia, Universidade do Minho, Braga, Portugal

Jennifer Cunha

Tânia nunes, ana rita nunes, tânia moreira, josé carlos núñez.

2 Departamento de Psicología, Universidad de Oviedo, Oviedo, Spain

Associated Data

Existing literature has analyzed homework characteristics associated with academic results. Researchers and educators defend the need to provide quality homework, but there is still much to be learned about the characteristics of quality homework (e.g., purposes, type). Acknowledging that teachers play an important role in designing and assigning homework, this study explored teachers’ perspectives regarding: (i) the characteristics of quality homework and (ii) the characteristics of the homework tasks assigned. In the current study, mathematics teachers from elementary and middle schools ( N = 78) participated in focus group discussions. To enhance the trustworthiness of the findings, homework tasks assigned by 25% of the participants were analyzed for triangulation of data purposes. Data were analyzed using thematic analysis for elementary and middle school separately. Teachers discussed the various characteristics of quality homework (e.g., short assignments, adjusted to the availability of students) and shared the characteristics of the homework tasks typically assigned, highlighting a few differences (e.g., degree of individualization of homework, purposes) between these two topics. Globally, data on the homework tasks assigned were consistent with teachers’ reports about the characteristics of the homework tasks they usually assigned. Findings provide valuable insights for research and practice aimed to promote the quality of homework and consequently students’ learning and progress.

Introduction

The extensive literature on homework suggests the importance of completing homework tasks to foster students’ academic achievement (e.g., Trautwein and Lüdtke, 2009 ; Hagger et al., 2015 ; Núñez et al., 2015a ; Valle et al., 2016 ; Fernández-Alonso et al., 2017 ). However, existing research also indicate that the amount of homework assigned is not always related to high academic achievement ( Epstein and Van Voorhis, 2001 ; Epstein and Van Voorhis, 2012 ). In the words of Dettmers et al. (2010) “homework works if quality is high” (p. 467). However, further research is needed to answer the question “What is quality homework?”.

Teachers are responsible for designing and assigning homework, thus our knowledge on their perspectives about this topic and the characteristics of the homework typically assigned is expected to be a relevant contribution to the literature on the quality of homework. Moreover, data on the characteristics of homework could provide valuable information to unveil the complex network of relationships between homework and academic achievement (e.g., Cooper, 2001 ; Trautwein and Köller, 2003 ; Trautwein et al., 2009a ; Xu, 2010 ).

Thus, focusing on the perspective of mathematics teachers from elementary and middle school, the aims of the present study are twofold: to explore the characteristics of quality homework, and to identify the characteristics of the homework tasks typically assigned at these school levels. Findings may help deepen our understanding of why homework may impact differently the mathematics achievement of elementary and middle school students (see Fan et al., 2017 ).

Research Background on Homework Characteristics

Homework is a complex educational process involving a diverse set of variables that each may influence students’ academic outcomes (e.g., Corno, 2000 ; Trautwein and Köller, 2003 ; Cooper et al., 2006 ; Epstein and Van Voorhis, 2012 ). Cooper (1989 , 2001 ) presented a model outlining the factors that may potentially influence the effect of homework at the three stages of the homework process (i.e., design of the homework assignment, completion of homework and homework follow-up practices). At the first stage teachers are expected to consider class characteristics (e.g., students’ prior knowledge, grade level, number of students per class), and also variables that may influence the impact of homework on students’ outcomes, such as homework assignment characteristics. In 1989, Cooper (see also Cooper et al., 2006 ) presented a list of the characteristics of homework assignments as follows: amount (comprising homework frequency and length), purpose, skill area targeted, degree of individualization, student degree of choice, completion deadlines, and social context. Based on existing literature, Trautwein et al. (2006b) proposed a distinct organization for the assignment characteristics. The proposal included: homework frequency (i.e., how often homework assignments are prescribed to students), quality, control, and adaptivity. “Homework frequency” and “adaptivity” are similar to “amount” and “degree of individualization” in Cooper’s model, respectively. Both homework models provide a relevant theoretical framework for the present study.

Prior research has analyzed the relationship between homework variables, students’ behaviors and academic achievement, and found different results depending on the variables examined (see Trautwein et al., 2009b ; Fan et al., 2017 ). For example, while homework frequency consistently and positively predicted students’ academic achievement (e.g., Trautwein et al., 2002 ; Trautwein, 2007 ; Fernández-Alonso et al., 2015 ), findings regarding the amount of homework assigned (usually assessed by the time spent on homework) have shown mixed results (e.g., Trautwein, 2007 ; Dettmers et al., 2009 ; Núñez et al., 2015a ). Data indicated a positive association between the amount of homework and students’ academic achievement in high school (e.g., OECD, 2014a ); however, this relationship is almost null in elementary school (e.g., Cooper et al., 2006 ; Rosário et al., 2009 ). Finally, other studies reported a negative association between time spent on homework and students’ academic achievement at different school levels (e.g., Trautwein et al., 2009b ; Rosário et al., 2011 ; Núñez et al., 2015a ).

Homework purposes are among the factors that may influence the effect of homework on students’ homework behaviors and academic achievement ( Cooper, 2001 ; Trautwein et al., 2009a ; Epstein and Van Voorhis, 2012 ; Rosário et al., 2015 ). In his model Cooper (1989 , 2001 ) reported instructional purposes (i.e., practicing or reviewing, preparation, integration and extension) and non-instructional purposes (i.e., parent-child communication, fulfilling directives, punishment, and community relations). Depending on their nature, homework instructional purposes may vary throughout schooling ( Muhlenbruck et al., 2000 ; Epstein and Van Voorhis, 2001 ). For example, in elementary school, teachers are likely to use homework as an opportunity to review the content taught in class, while in secondary school (6th–12th grade), teachers are prone to use homework to prepare students for the content to be learned in subsequent classes ( Muhlenbruck et al., 2000 ). Still, studies have recently shown that practicing the content learned is the homework purpose most frequently used throughout schooling (e.g., Xu and Yuan, 2003 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ). Studies using quantitative methodologies have analyzed the role played by homework purposes in students’ effort and achievement ( Trautwein et al., 2009a ; Rosário et al., 2015 , 2018 ), and reported distinct results depending on the subject analyzed. For example, Foyle et al. (1990) found that homework assignments with the purposes of practice and preparation improved the performance of 5th-grade students’ social studies when compared with the no-homework group. However, no statistical difference was found between the two types of homework purposes analyzed (i.e., practice and preparation). When examining the homework purposes reported by 8th-grade teachers of French as a Second Language (e.g., drilling and practicing, motivating, linking school and home), Trautwein et al. (2009a) found that students in classes assigned tasks with high emphasis on motivation displayed more effort and achieved higher outcomes than their peers. On the contrary, students in classes assigned tasks with high drill and practice reported less homework effort and achievement ( Trautwein et al., 2009a ). A recent study by Rosário et al. (2015) analyzed the relationship between homework assignments with various types of purposes (i.e., practice, preparation and extension) and 6th-grade mathematics achievement. These authors reported that homework with the purpose of “extension” impacted positively on students’ academic achievement while the other two homework purposes did not.

Cooper (1989 , 2001 ) identified the “degree of individualization” as a characteristic of homework focused on the need to design homework addressing different levels of performance. For example, some students need to be assigned practice exercises with a low level of difficulty to help them reach school goals, while others need to be assigned exercises with high levels of complexity to foster their motivation for homework ( Trautwein et al., 2002 ). When there is a disparity between the level of difficulty of homework assignments and students’ skills level, students may have to spend long hours doing homework, and they may experience negative emotions or even avoid doing homework ( Corno, 2000 ). On the contrary, when homework assignments meet students’ learning needs (e.g., Bang, 2012 ; Kukliansky et al., 2014 ), both students’ homework effort and academic achievement increase (e.g., Trautwein et al., 2006a ; Zakharov et al., 2014 ). Teachers may also decide on the time given to students to complete their homework ( Cooper, 1989 ; Cooper et al., 2006 ). For example, homework may be assigned to be delivered in the following class (e.g., Kaur et al., 2004 ) or within a week (e.g., Kaur, 2011 ). However, research on the beneficial effects of each practice is still limited.

Trautwein et al. (2006b) investigated homework characteristics other than those previously reported. Their line of research analyzed students’ perception of homework quality and homework control (e.g., Trautwein et al., 2006b ; Dettmers et al., 2010 ). Findings on homework quality (e.g., level of difficulty of the mathematics exercises, Trautwein et al., 2002 ; homework “cognitively activating” and “well prepared”, Trautwein et al., 2006b , p. 448; homework selection and level of challenge, Dettmers et al., 2010 ; Rosário et al., 2018 ) varied regarding the various measures and levels of analysis considered. For example, focusing on mathematics, Trautwein et al. (2002) concluded that “demanding” exercises improved 7th-grade students’ achievement at student and class levels, while “repetitive exercises” impacted negatively on students’ achievement. Dettmers et al. (2010) found that homework assignments perceived by students as “well-prepared and interesting” (p. 471) positively predicted 9th- and 10th-grade students’ homework motivation (expectancy and value beliefs) and behavior (effort and time) at student and class level, and mathematics achievement at class level only. These authors also reported that “cognitively challenging” homework (p. 471), as perceived by students, negatively predicted students’ expectancy beliefs at both levels, and students’ homework effort at student level ( Dettmers et al., 2010 ). Moreover, this study showed that “challenging homework” significantly and positively impacted on students’ mathematics achievement at class level ( Dettmers et al., 2010 ). At elementary school, homework quality (assessed through homework selection) predicted positively 6th-grade students’ homework effort, homework performance, and mathematics achievement ( Rosário et al., 2018 ).

Finally, Trautwein and colleagues investigated the variable “homework control” perceived by middle school students and found mixed results. The works by Trautwein and Lüdtke (2007 , 2009 ) found that “homework control” predicted positively students’ homework effort in mathematics, but other studies (e.g., Trautwein et al., 2002 , 2006b ) did not predict homework effort and mathematics achievement.

The Present Study

A vast body of research indicates that homework enhances students’ academic achievement [see the meta-analysis conducted by Fan et al. (2017) ], however, maladaptive homework behaviors of students (e.g., procrastination, lack of interest in homework, failure to complete homework) may affect homework benefits ( Bembenutty, 2011a ; Hong et al., 2011 ; Rosário et al., 2019 ). These behaviors may be related to the characteristics of the homework assigned (e.g., large amount of homework, disconnect between the type and level of difficulty of homework assignments and students’ needs and abilities, see Margolis and McCabe, 2004 ; Trautwein, 2007 ).

Homework is only valuable to students’ learning when its quality is perceived by students ( Dettmers et al., 2010 ). Nevertheless, little is known about the meaning of homework quality for teachers who are responsible for assigning homework. What do teachers understand to be quality homework? To our knowledge, the previous studies exploring teachers’ perspectives on their homework practices did not relate data with quality homework (e.g., Xu and Yuan, 2003 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ). For example, Kukliansky et al. (2014) found a disconnect between middle school science teachers’ perspectives about their homework practices and their actual homework practices observed in class. However, results were not further explained.

The current study aims to explore teachers’ perspectives on the characteristics of quality homework, and on the characteristics underlying the homework tasks assigned. Findings are expected to shed some light on the role of teachers in the homework process and contribute to maximize the benefits of homework. Our results may be useful for either homework research (e.g., by informing new quantitative studies grounded on data from teachers’ perspectives) or educational practice (e.g., by identifying new avenues for teacher training and the defining of guidelines for homework practices).

This study is particularly important in mathematics for the following reasons: mathematics is among the school subjects where teachers assign the largest amount of homework (e.g., Rønning, 2011 ; Xu, 2015 ), while students continue to yield worrying school results in the subject, especially in middle and high school ( Gottfried et al., 2007 ; OECD, 2014b ). Moreover, a recent meta-analysis focused on mathematics and science homework showed that the relationship between homework and academic achievement in middle school is weaker than in elementary school ( Fan et al., 2017 ). Thus, we collected data through focus group discussions with elementary and middle school mathematics teachers in order to analyze any potential variations in their perspectives on the characteristics of quality homework, and on the characteristics of homework tasks they typically assign. Regarding the latter topic, we also collected photos of homework tasks assigned by 25% of the participating teachers in order to triangulate data and enhance the trustworthiness of our findings.

Our exploratory study was guided by the following research questions:

simple (1) How do elementary and middle school mathematics teachers perceive quality homework?
simple (2) How do elementary and middle school mathematics teachers describe the homework tasks they typically assign to students?

Materials and Methods

The study context.

Despite recommendations of the need for clear homework policies (e.g., Cooper et al., 2006 ; Bembenutty, 2011b ), Portugal has no formal guidelines for homework (e.g., concerning the frequency, length, type of tasks). Still, many teachers usually include homework as part of students’ overall grade and ask parents to monitor their children’s homework completion. Moreover, according to participants there is no specific training on homework practices for pre-service or in-service teachers.

The Portuguese educational system is organized as follows: the last two years of elementary school encompass 5th and 6th grade (10 and 11 years old), while middle school encompasses 7th, 8th, and 9th grade (12 to 14 years old). At the two school levels mentioned, mathematics is a compulsory subject and students attend three to five mathematics lessons per week depending on the duration of each class (270 min per week for Grades 5 and 6, and 225 min per week for Grades 7–9). All students are assessed by their mathematics teacher (through continuous assessment tests), and at the end of elementary and middle school levels (6th and 9th grade) students are assessed externally through a national exam that counts for 30% of the overall grade. In Portuguese schools assigning homework is a frequently used educational practice, mostly in mathematics, and usually counts toward the overall grade, ranging between 2% and 5% depending on school boards ( Rosário et al., 2018 ).

Participants

In the current study, all participants were involved in focus groups and 25% of them, randomly selected, were asked to submit photos of homework tasks assigned.

According to Morgan (1997) , to maximize the discussion among participants it is important that they share some characteristics and experiences related to the aims of the study in question. In the current study, teachers were eligible to participate when the following criteria were met: (i) they had been teaching mathematics at elementary or middle school levels for at least two years; and (ii) they would assign homework regularly, at least twice a week, in order to have enough experiences to share in the focus group.

All mathematics teachers ( N = 130) from 25 elementary and middle schools in Northern Portugal were contacted by email. The email informed teachers of the purposes and procedures of the study (e.g., inclusion criteria, duration of the session, session videotaping, selection of teachers to send photos of homework tasks assigned), and invited them to participate in the study. To facilitate recruitment, researchers scheduled focus group discussions considering participants’ availability. Of the volunteer teachers, all participants met the inclusion criteria. The research team did not allocate teachers with hierarchical relationships in the same group, as this might limit freedom of responses, affect the dynamics of the discussion, and, consequently, the outcomes ( Kitzinger, 1995 ).

Initially we conducted four focus groups with elementary school teachers (5th and 6th grade, 10 and 11 years old) and four focus groups with middle school teachers (7th, 8th, and 9th grade, 12, 13 and 14 years old). Subsequently, two additional focus group discussions (one for each school level) were conducted to ensure the saturation of data. Finally, seventy-eight mathematics teachers (61 females and 17 males; an acceptance rate of 60%) from 16 schools participated in our study (see Table 1 ). The teachers enrolled in 10 focus groups comprised of seven to nine teachers per group. Twenty teachers were randomly selected and asked to participate in the second data collection; all answered positively to our invitation (15 females and 5 males).

Participants’ demographic information.

According to our participants, in the school context, mathematics teachers may teach one to eight classes of different grade levels. In the current research, participants were teaching one to five classes of two or three grade levels at schools in urban or near urban contexts. The participants practiced the mandatory nationwide curriculum and a continuous assessment policy.

Data Collection

We carried out this study following the recommendations of the ethics committee of the University of Minho. All teachers gave written informed consent to participate in the research in accordance with the Declaration of Helsinki. The collaboration involved participating in one focus group discussion, and, for 25% of the participants, submitting photos by email of the homework tasks assigned.

In the current study, aiming to deepen our comprehension of the research questions, focus group interviews were conducted to capture participants’ thoughts about a particular topic ( Kitzinger, 1995 ; Morgan, 1997 ). The focus groups were conducted by two members of the research team (a moderator and a field note-taker) in the first term of the school year and followed the procedure described by Krueger and Casey (2000) . To prevent mishandling the discussions and to encourage teachers to participate in the sessions, the two facilitators attended a course on qualitative research offered at their home institution specifically targeting focus group methodology.

All focus group interviews were videotaped. The sessions were held in a meeting room at the University of Minho facilities, and lasted 90 to 105 min. Before starting the discussion, teachers filled in a questionnaire with sociodemographic information, and were invited to read and sign a written informed consent form. Researchers introduced themselves, and read out the information regarding the study purpose and the focus group ground rules. Participants were ensured of the confidentiality of their responses (e.g., names and researchers’ personal notes that might link participants to their schools were deleted). Then, the investigators initiated the discussion (see Table 2 ). At the end of each focus group discussion, participants were given the opportunity to ask questions or make further contributions.

Focus group questions.

After the focus group discussions, we randomly selected 25% of the participating teachers (i.e., 10 teachers from each school level), each asked to submit photos of the homework tasks assigned by email over the course of three weeks (period between two mathematics assessment tests). This data collection aimed to triangulate data from focus groups regarding the characteristics of homework usually assigned. To encourage participation, the research team sent teachers a friendly reminder email every evening throughout the period of data collection. In total, we received 125 photos (51% were from middle school teachers).

Data Analysis

Videotapes were used to assist the verbatim transcription of focus group data. Both focus group data and photos of the homework assignments were analyzed using thematic analysis ( Braun and Clarke, 2006 ), assisted by QSR International’s NVivo 10 software ( Richards, 2005 ). In this analysis there are no rigid guidelines on how to determine themes; to assure that the analysis is rigorous, researchers are expected to follow a consistent procedure throughout the analysis process ( Braun and Clarke, 2006 ). For the current study, to identify themes and sub-themes, we used the extensiveness of comments criterion (number of participants who express a theme, Krueger and Casey, 2000 ).

Firstly, following an inductive process one member of the research team read the first eight focus group transcriptions several times, took notes on the overall ideas of the data, and made a list of possible codes for data at a semantic level ( Braun and Clarke, 2006 ). Using a cluster analysis by word similarity procedure in Nvivo, all codes were grouped in order to identify sub-themes and themes posteriorly. All the themes and sub-themes were independently and iteratively identified and compared with the literature on homework ( Peterson and Irving, 2008 ). Then, the themes and sub-themes were compared with the homework characteristics already reported in the literature (e.g., Cooper, 1989 ; Epstein and Van Voorhis, 2001 ; Trautwein et al., 2006b ). New sub-themes emerged from participants’ discourses (i.e., “adjusted to the availability of students,” “teachers diagnose learning”), and were grouped in the themes reported in the literature. After, all themes and sub-themes were organized in a coding scheme (for an example see Table 3 ). Finally, the researcher coded the two other focus group discussions, no new information was added related to the research questions. Given that the generated patterns of data were not changed, the researcher concluded that thematic saturation was reached.

Examples of the coding scheme.

An external auditor, trained on the coding scheme, revised all transcriptions, the coding scheme and the coding process in order to minimize researchers’ biases and increase the trustworthiness of the study ( Lincoln and Guba, 1985 ). The first author and the external auditor examined the final categorization of data and reached consensus.

Two other members of the research team coded independently the photos of the homework assignments using the same coding scheme of the focus groups. To analyze data, the researchers had to define the sub-themes “short assignments” (i.e., up to three exercises) and “long assignments” (i.e., more than three exercises). In the end, the two researchers reviewed the coding process and discussed the differences found (e.g., some exercises had several sub questions, so one of the researchers coded it as “long assignments”; see the homework sample 4 of the Supplementary Material ). However, the researchers reached consensus, deciding not to count the number of sub questions of each exercise individually, because these types of questions are related and do not require a significant amount of additional time.

Inter-rater reliability (Cohen’s Kappa) was calculated. The Cohen’s Kappa was 0.86 for the data analysis of the focus groups and 0.85 for data analysis of the photos of homework assignments, which is considered very good according to Landis and Koch (1977) . To obtain a pattern of data considering the school levels, a matrix coding query was run for each data source (i.e., focus groups and photos of homework assignments). Using the various criteria options in NVivo 10, we crossed participants’ classifications (i.e., school level attribute) and nodes and displayed the frequencies of responses for each row–column combination ( Bazeley and Jackson, 2013 ).

In the end of this process of data analysis, for establishing the trustworthiness of findings, 20 teachers (i.e., ten participants of each grade level) were randomly invited, and all agreed, to provide a member check of the findings ( Lincoln and Guba, 1985 ). Member checking involved two phases. First, teachers were asked individually to read a summary of the findings and to fill in a 5-point Likert scale (1, completely disagree; 5, completely agree) with four items: “Findings reflect my perspective regarding homework quality”; “Findings reflect my perspective regarding homework practices”; “Findings reflect what was discussed in the focus group where I participated”, and “I feel that my opinion was influenced by the other teachers during the discussion” (inverted item). Secondly, teachers were gathered by school level and asked to critically analyze and discuss whether an authentic representation was made of their perspectives regarding quality homework and homework practices ( Creswell, 2007 ).

This study explored teachers’ perspectives on the characteristics of quality homework, and on the characteristics of the homework tasks typically assigned. To report results, we used the frequency of occurrence criterion of the categories defined by Hill et al. (2005) . Each theme may be classified as “General” when all participants, or all except one, mention a particular theme; “Typical” when more than half of the cases mention a theme; “Variant” when more than 3, and less than half of the cases mention a theme; and “Rare” when the frequency is between 2 and 3 cases. In the current study, only general and typical themes were reported to discuss the most salient data.

The results section was organized by each research question. Throughout the analysis of the results, quotes from participants were presented to illustrate data. For the second research question, data from the homework assignments collected as photographs were also included.

Initial Data Screening

All participating teachers defended the importance of completing homework, arguing that homework can help students to develop their learning and to engage in school life. Furthermore, participants also agreed on the importance of delivering this message to students. Nevertheless, all teachers acknowledged that assigning homework daily present a challenge to their teaching routine because of the heavy workload faced daily (e.g., large numbers of students per class, too many classes to teach, teaching classes from different grade levels which means preparing different lessons, administrative workload).

Teachers at both school levels talked spontaneously about the nature of the tasks they usually assign, and the majority reported selecting homework tasks from a textbook. However, participants also referred to creating exercises fit to particular learning goals. Data collected from the homework assigned corroborated this information. Most of participating teachers reported that they had not received any guidance from their school board regarding homework.

How do Elementary and Middle School Teachers Perceive Quality Homework?

Three main themes were identified by elementary school teachers (i.e., instructional purposes, degree of individualization/adaptivity, and length of homework) and two were identified by middle school teachers (i.e., instructional purposes, and degree of individualization/adaptivity). Figure 1 depicts the themes and sub-themes reported by teachers in the focus groups.

An external file that holds a picture, illustration, etc.
Object name is fpsyg-10-00224-g001.jpg

Characteristics of quality homework reported by mathematics teachers by school level.

In all focus group discussions, all teachers from elementary and middle school mentioned “instructional purposes” as the main characteristic of quality homework. When asked to further explain the importance of this characteristic, teachers at both school levels in all focus group talked about the need for “practicing or reviewing” the content delivered in class to strengthen students’ knowledge. A teacher illustrated this idea clearly: “it is not worth teaching new content when students do not master the material previously covered” (P1 FG3). This idea was supported by participants in all focus groups; “at home they [students] have to work on the same content as those taught in class” (P1 FG7), “students have to revisit exercises and practice” (P2 FG9), “train over and over again” (P6 FG1), “practice, practice, practice” (P4 FG2).

While discussing the benefits of designing homework with the purpose of practicing the content learned, teachers at both school levels agreed on the fact that homework may be a useful tool for students to diagnose their own learning achievements while working independently. Teachers were empathetic with their peers when discussing the instrumentality of homework as a “thermometer” for students to assess their own progress. This idea was discussed in similar ways in all focus group, as the following quotation illustrates:

P2 FG1: Homework should be a bridge between class and home… students are expected to work independently, learn about their difficulties when doing homework, and check whether they understood the content.

When asked to outline other characteristics of quality homework, several elementary school teachers in all focus group mentioned that quality homework should also promote “student development” as an instructional purpose. These participants explained that homework is an instructional tool that should be designed to “foster students’ autonomy” (P9 FG4), “develop study habits and routines” (P1 FG8), and “promote organization skills and study methods” (P6 FG7). These thoughts were unanimous among participants in all focus groups. While some teachers introduced real-life examples to illustrate the ideas posited by their colleagues, others nodded their heads in agreement.

In addition, some elementary school teachers observed that homework tasks requiring transference of knowledge could help develop students’ complex thinking, a highly valued topic in the current mathematics curriculum worldwide. Teachers discussed this topic enthusiastically in two opposite directions: while some teachers defended this purpose as a characteristic of quality homework, others disagreed, as the following conversation excerpt illustrates:

P7 FG5: For me good homework would be a real challenge, like a problem-solving scenario that stimulates learning transference and develops mathematical reasoning … mathematical insight. It’s hard because it forces them [students] to think in more complex ways; still, I believe this is the type of homework with the most potential gains for them.

P3 FG5: That’s a good point, but they [students] give up easily. They just don’t do their homework. This type of homework implies competencies that the majority of students do not master…

P1 FG5: Not to mention that this type of homework takes up a lot of teaching time… explaining, checking…, and we simply don’t have time for this.

Globally, participants agreed on the potential of assigning homework with the purpose of instigating students to transfer learning to new tasks. However, participants also discussed the limitations faced daily in their teaching (e.g., number of students per class, students’ lack of prior knowledge) and concluded that homework with this purpose hinders the successful development of their lesson plans. This perspective may help explain why many participants did not perceive this purpose as a significant characteristic of quality homework. Further commenting on the characteristics of quality homework, the majority of participants at both school levels agreed that quality homework should be tailored to meet students’ learning needs. The importance of individualized homework was intensely discussed in all focus groups, and several participants suggested the need for designing homework targeted at a particular student or groups of students with common education needs. The following statements exemplifies participants’ opinions:

P3 FG3: Ideally, homework should be targeted at each student individually. For André a simple exercise, for Ana a more challenging exercise … in an ideal world homework should be tailored to students’ needs.

P6 FG6: Given the diversity of students in our classes, we may find a rainbow of levels of prior knowledge… quality homework should be as varied as our students’ needs.

As discussed in the focus groups, to foster the engagement of high-achievers in homework completion, homework tasks should be challenging enough (as reported previously by P3 FG3). However, participants at both school levels observed that their heavy daily workload prevents them from assigning individualized homework:

P1 FG1: I know it’s important to assign differentiated homework tasks, and I believe in it… but this option faces real-life barriers, such as the number of classes we have to teach, each with thirty students, tons of bureaucratic stuff we have to deal with… All this raises real-life questions, real impediments… how can we design homework tasks for individual students?

Considering this challenge, teachers from both school levels suggested that quality homework should comprise exercises with increasing levels of difficulty. This strategy would respond to the heterogeneity of students’ learning needs without assigning individualized homework tasks to each student.

While discussing individualized homework, elementary school teachers added that assignments should be designed bearing in mind students’ availability (e.g., school timetable, extracurricular activities, and exam dates). Participants noted that teachers should learn the amount of workload their students have, and should be aware about the importance of students’ well-being.

P4 FG1: If students have large amounts of homework, this could be very uncomfortable and even frustrating… They have to do homework of other subjects and add time to extracurricular activities… responding to all demands can be very stressful.

P4 FG2: I think that we have to learn about the learning context of our students, namely their limitations to complete homework in the time they have available. We all have good intentions and want them to progress, but if students do not have enough time to do their homework, this won’t work. So, quality homework would be, for example, when students have exams and the teacher gives them little or no homework at all.

The discussion about the length of homework found consensus among the elementary school teachers in all focus group in that quality homework should be “brief”. During the discussions, elementary school teachers further explained that assigning long tasks is not beneficial because “they [students] end up demotivated” (P3 FG4). Besides, “completing long homework assignments takes hours!” (P5 FG4).

How do Elementary and Middle School Teachers Describe the Homework Tasks They Typically Assign to Students?

When discussing the characteristics of the homework tasks usually assigned to their students four main themes were identified by elementary school teachers (i.e., instructional purposes, degree of individualization/adaptivity, frequency and completion deadlines), and two main themes were raised by middle school (i.e., instructional purposes, and degree of individualization/adaptivity). Figure 2 gives a general overview of the findings. Data gathered from photos added themes to findings as follows: one (i.e., length) to elementary school and two (i.e., length and completion deadlines) to middle school (see Figure 3 ).

An external file that holds a picture, illustration, etc.
Object name is fpsyg-10-00224-g002.jpg

Characteristics of the homework tasks usually assigned as reported by mathematics teachers.

An external file that holds a picture, illustration, etc.
Object name is fpsyg-10-00224-g003.jpg

Characteristics of the homework tasks assigned by mathematics teachers.

While describing the characteristics of the homework tasks usually assigned, teachers frequently felt the need to compare the quality homework characteristics previously discussed with those practices. In fact, at this stage, teachers’ discourse was often focused on the analysis of the similarities and potential discrepancies found.

The majority of teachers at both school levels in all focus group reported that they assign homework with the purpose of practicing and reviewing the materials covered earlier. Participants at both school levels highlighted the need to practice the contents covered because by the end of 6th- and 9th-grade students have to sit for a national exam for which they have to be trained. This educational context may interfere with the underlying homework purposes teachers have, as this quotation illustrates:

P3 FG3: When teaching mathematics, we set several goals, but our main focus is always the final exam they [students] have to take. I like students who think for themselves, who push themselves out of their comfort zone. However, I’m aware that they have to score high on national exams, otherwise… so, I assign homework to practice the contents covered.

Beyond assigning homework with the purpose of practicing and reviewing, middle school teachers also mentioned assigning homework with the purpose of diagnosing skills and personal development (see Figure 2 ). Many teachers reported that they use homework as a tool to diagnose students’ skills. However, several recognized that they had previously defended the importance of homework to help students to evaluate their own learning (see Figure 1 ). When discussing the latter point, participants observed the need to find out about whether students had understood the content taught in class, and to decide which changes to teaching style, homework assigned, or both may be necessary.

Participant teachers at middle school in all focus groups profusely discussed the purpose of personal development when assigning homework. In fact, not many teachers at this school level mentioned this purpose as a characteristic of quality homework (it was a variant category, so it was not reported), yet it was referred to as a cornerstone in their homework practice. Reflecting on this discrepancy, middle school teachers explained in a displeased tone that their students were expected to have developed study habits and manage their school work with autonomy and responsibility. However, this “educational scenario is rare, so I feel the need to assign homework with this aim [personal development]” (P4 FG9).

Moving further in the discussion, the majority of teachers at both school levels reported to assign whole-class homework (homework designed for the whole class with no focus on special cases). “Individualized homework requires a great amount of time to be monitored” (P1 FG6), explained several participants while recalling earlier comments. Teachers justified their position referring to the impediments already mentioned (e.g., large number of students per class, number of classes from different grade levels which means preparing different lessons). Besides, teachers discussed the challenge of coping with heterogeneous classes, as one participant noted: “the class is so diverse that it is difficult to select homework tasks to address the needs of every single student. I would like to do it…but we do not live in an ideal world” (P9 FG4).

Moreover, teachers at both school levels (see Figure 2 ) reported to assign homework according to the availability of students; still, only elementary school teachers had earlier referred to the importance of this characteristic in quality homework. When teachers were asked to elaborate on this idea, they defended the need to negotiate with students about specific homework characteristics, for example, the amount of homework and submission deadline. In some classes, matching students’ requests, teachers might assign a “weekly homework pack” (P7 FG10). This option provides students with the opportunity to complete homework according to their availability (e.g., choosing some days during the week or weekend). Teachers agreed that ‘negotiation’ fosters students’ engagement and homework compliance (e.g., “I do not agree that students do homework on weekends, but if they show their wish and actually they complete it, for me that’s okay”, P7 FG10). In addition, teachers expressed worry about their students’ often heavy workload. Many students stay in school from 8.30 am to 6.30 pm and then attend extracurricular activities (e.g., soccer training, private music lessons). These activities leave students very little free time to enjoy as they wish, as the following statement suggests:

P8 FG4: Today I talked to a group of 5th-graders which play soccer after school three times a week. They told me that sometimes they study between 10.00 and 11.00 p.m. I was astonished. How is this possible? It’s clearly too much for these kids.

Finally, elementary school teachers in all focus group referred frequency and completion deadlines as characteristics of the homework they usually assign. The majority of teachers informed that they assign homework in almost every class (i.e., teachers reported to exclude tests eves of other subjects), to be handed in the following class.

The photos of the homework assignments (see some examples in Supplementary Material ) submitted by the participating teachers served to triangulate data. The analysis showed that teachers’ discourses about the characteristics of homework assigned and the homework samples are congruent, and added information about the length of homework (elementary and middle schools) and the completion deadlines (middle school) (see Figure 3 ).

Discussion and Implications for Practice and Research

Homework research have reported teachers’ perspectives on their homework practices (e.g., Brock et al., 2007 ; Danielson et al., 2011 ; Kaur, 2011 ; Bang, 2012 ; Kukliansky et al., 2014 ), however, literature lacks research on the quality of homework. This study adds to the literature by examining the perspectives of teachers from two school levels regarding quality homework. Moreover, participants described the characteristics of the homework assignments they typically assign, which triggered the discussion about the match between the characteristics of quality homework and the tasks actually assigned. While discussing these key aspects of the homework process, the current study provides valuable information which may help deepen our understanding of the different contributions of homework to students’ learning. Furthermore, findings are expected to inform teachers and school administrators’ homework practices and, hopefully, improve the quality of students’ learning.

All teachers at both school levels valued homework as an important educational tool for their teaching practice. Consistent with the literature, participants indicated practicing or reviewing the material covered in class as the main purpose of both the homework typically assigned ( Danielson et al., 2011 ; Kaur, 2011 ) and quality homework. Despite the extended use of this homework purpose by teachers, a recent study conducted with mathematics teachers found that homework with the purpose of practicing the material covered in class did not impact significantly the academic achievement of 6th-grade students; however, homework designed with the purpose of solving problems did (extension homework) ( Rosário et al., 2015 ). Interestingly, in the current study only teachers from elementary school mentioned the homework purpose “extension” as being part of quality homework, but these teachers did not report to use it in practice (at least it was not a typical category) (see Figure 2 ). Extension homework was not referenced by middle school teachers either as quality homework or as a characteristic of homework assigned. Given that middle school students are expected to master complex math skills at this level (e.g., National Research Council and Mathematics Learning Study Committee, 2001 ), this finding may help school administrators and teachers reflect on the value and benefits of homework to students learning progress.

Moreover, teachers at both school levels stressed the use of homework as a tool to help students evaluate their own learning as a characteristic of quality homework; however, this purpose was not said to be a characteristic of the homework usually assigned. If teachers do not explicitly emphasize this homework purpose to their students, they may not perceive its importance and lose opportunities to evaluate and improve their work.

In addition, elementary school teachers identified personal development as a characteristic of quality homework. However, only middle school teachers reported assigning homework aiming to promote students’ personal development, and evaluate students’ learning (which does not imply that students evaluate their own learning). These findings are important because existing literature has highlighted the role played by homework in promoting students’ autonomy and learning throughout schooling ( Rosário et al., 2009 , 2011 ; Ramdass and Zimmerman, 2011 ; Núñez et al., 2015b ).

Globally, data show a disconnect between what teachers believe to be the characteristics of quality homework and the characteristics of the homework assigned, which should be further analyzed in depth. For example, teachers reported that middle school students lack the autonomy and responsibility expected for this school level, which translates to poor homework behaviors. In fact, contrary to what they would expect, middle school teachers reported the need to promote students’ personal development (i.e., responsibility and autonomy). This finding is consistent with the decrease of students’ engagement in academic activities found in middle school (e.g., Cleary and Chen, 2009 ; Wang and Eccles, 2012 ). This scenario may present a dilemma to middle school teachers regarding the purposes of homework. On one hand, students should have homework with more demanding purposes (e.g., extension); on another hand, students need to master work habits, responsibility and autonomy, otherwise homework may be counterproductive according to the participating teachers’ perspective.

Additionally, prior research has indicated that classes assigned challenging homework demonstrated high mathematics achievement ( Trautwein et al., 2002 ; Dettmers et al., 2010 ). Moreover, the study by Zakharov et al. (2014) found that Russian high school students from basic and advanced tracks benefited differently from two types of homework (i.e., basic short-answer questions, and open-ended questions with high level of complexity). Results showed that a high proportion of basic or complex homework exercises enhanced mathematics exam performance for students in the basic track; whereas only a high proportion of complex homework exercises enhanced mathematics exam performance for students in the advanced track. In fact, for these students, a low proportion of complex homework exercises was detrimental to their achievement. These findings, together with our own, may help explain why the relationship between homework and mathematics achievement in middle school is lower than in elementary school (see Fan et al., 2017 ). Our findings suggest the need for teachers to reflect upon the importance of assigning homework to promote students’ development in elementary school, and of assigning homework with challenging purposes as students advance in schooling to foster high academic outcomes. There is evidence that even students with poor prior knowledge need assignments with some degree of difficulty to promote their achievement (see Zakharov et al., 2014 ). It is important to note, however, the need to support the autonomy of students (e.g., providing different the types of assignments, opportunities for students to express negative feelings toward tasks, answer students’ questions) to minimize the threat that difficult homework exercises may pose to students’ sense of competence; otherwise an excessively high degree of difficulty can lead to students’ disengagement (see Patall et al., 2018 ). Moreover, teachers should consider students’ interests (e.g., which contents and types of homework tasks students like) and discuss homework purposes with their students to foster their understanding of the tasks assigned and, consequently, their engagement in homework ( Xu, 2010 , 2018 ; Epstein and Van Voorhis, 2012 ; Rosário et al., 2018 ).

We also found differences between teachers’ perspectives of quality homework and their reported homework practices concerning the degree of individualization when assigning homework. Contrary to the perspectives that quality homework stresses individual needs, teachers reported to assign homework to the whole class. In spite of the educational costs associated with assigning homework adjusted to specific students or groups of students (mentioned several times by participants), research has reported benefits for students when homework assignments match their educational needs (e.g., Cooper, 2001 ; Trautwein et al., 2006a ; Zakharov et al., 2014 ). The above-mentioned study by Zakharov et al. (2014) also shed light on this topic while supporting our participants’ suggestion to assign homework with increasing level of difficulty aiming to match the variety of students’ levels of knowledge (see also Dettmers et al., 2010 ). However, teachers did not mention this idea when discussing the characteristic of homework typically assigned. Thus, school administrators may wish to consider training teachers (e.g., using mentoring, see Núñez et al., 2013 ) to help them overcome some of the obstacles faced when designing and assigning homework targeting students’ individual characteristics and learning needs.

Another interesting finding is related to the sub-theme of homework adjusted to the availability of students. This was reported while discussing homework quality (elementary school) and characteristics of homework typically assigned (elementary and middle school). Moreover, some elementary and middle school teachers explained by email the reasons why they did not assign homework in some circumstances [e.g., eves of assessment tests of other subjects, extracurricular activities, short time between classes (last class of the day and next class in the following morning)]. These teachers’ behaviors show concern for students’ well-being, which may positively influence the relationship between students and teachers. As some participants mentioned, “students value this attitude” (P1 FG5). Thus, future research may explore how homework adjusted to the availability of students may contribute to encouraging positive behaviors, emotions and outcomes of students toward their homework.

Data gathered from the photos of the assigned homework tasks allowed a detailed analysis of the length and completion deadlines of homework. Long assignments did not match elementary school teachers’ perspectives of quality homework. However, a long homework was assigned once and aimed to help students practice the material covered for the mathematics assessment test. Here, practices diverged. Some teachers assigned this homework some weeks before and others assign it in last class before the test. For this reason, the “long term” completion deadline was not a typical category, hence not reported. Future research could consider studying the impact of this homework characteristic on students’ behaviors and academic performance.

Finally, our findings show that quality homework, according to teachers’ perspectives, requires attention to a combination of several characteristics of homework. Future studies may include measures to assess characteristics of homework other than “challenge” and “selection” already investigated ( Trautwein et al., 2006b ; Dettmers et al., 2010 ; Rosário et al., 2018 ); for example, homework adjusted to the availability of students.

Strengths and Limitations of the Study

The current study analyzed the teachers’ perspectives on the characteristics of quality homework and of the homework they typically assigned. Despite the incapability to generalize data, we believe that these findings provide important insights into the characteristics that may impact a homework assignment’s effectiveness, especially at middle school level. For example, our results showed a disconnect between teachers’ perspectives about the characteristics of quality homework and the characteristics of the homework they assign. This finding is relevant and emphasizes the need to reflect on the consistency between educational discourses and educational practices. Teachers and school administrators could consider finding opportunities to reflect on this disconnect, which may also occur in other educational practices (e.g., teacher feedback, types of questions asked in class). Present data indicate that middle school teachers reported to assign homework with the major purpose of practicing and reviewing the material, but they also aim to develop students’ responsibility and autonomy; still they neglect homework with the purpose of extension which is focused on encouraging students to display an autonomous role, solve problems and transfer the contents learned (see discussion section). Current findings also highlight the challenges and dilemmas teachers face when they assign homework, which is important to address in teachers’ training. In fact, assigning quality homework, that is, homework that works, is not an easy task for teachers and our findings provide empirical data to discuss and reflect upon its implications for research and educational practice. Although our findings cannot be generalized, still they are expected to provide important clues to enhance teachers’ homework practices in different contexts and educational settings, given that homework is among the most universal educational practices in the classroom, is a topic of public debate (e.g., some arguments against homework are related to the characteristics of the assignments, and to the malpractices in using this educational tool) and an active area of research in many countries ( Fan et al., 2017 ).

Moreover, these findings have identified some of the most common obstacles teachers struggle with; such data may be useful to school administrators when designing policies and to teacher training. The administrative obstacles (e.g., large number of students per class) reported by teachers may help understand some of the discrepancies found between teachers’ definition of quality homework and their actual homework practices (e.g., degree of individualization), and also identify which problems related to homework may require intervention. Furthermore, future research could further investigate this topic by interviewing teachers, videotaping classroom activities and discussing data in order to design new avenues of homework practices.

We share the perspective of Trautwein et al. (2006b) on the importance of mapping the characteristics of homework positively associated with students’ homework behaviors. Data from this study may inform future studies analyzing these relationships, promote adaptive homework behaviors and enhance learning.

Methodologically, this research followed rigorous procedures to increase the trustworthiness of findings, improving the validity of the study (e.g., Lincoln and Guba, 1985 ) that should be accounted for. Data from two data sources (i.e., focus groups and the homework assignments photographed) were consistent, and the member checking conducted in both phases allowed the opportunity to learn that the findings of the focus group seem to accurately reflect the overall teachers’ perspectives regarding quality homework and their homework practices.

Despite the promising contributions of this study to the body of research regarding homework practices, this specific research provides an incomplete perspective of the homework process as it has only addressed the perspectives of one of the agents involved. Future research may consider analyzing students’ perspectives about the same topic and contrast data with those of teachers. Findings are expected to help us identify the homework characteristics most highly valued by students and learn about whether they match those of teachers.

Furthermore, data from homework assignments (photos) were provided by 25% of the participating teachers and for a short period of time (i.e., three weeks in one school term). Future research may consider conducting small-scale studies by collecting data from various sources of information aiming at triangulating data (e.g., analyzing homework assignments given in class, interviewing students, conducting in-class observations) at different times of the school year. Researchers should also consider conducting similar studies in different subjects to compare data and inform teachers’ training.

Finally, our participants’ description does not include data regarding the teaching methodology followed by teachers in class. However, due to the potential interference of this variable in results, future research may consider collect and report data regarding school modality and the teaching methodology followed in class.

Homework is an instructional tool that has proved to enhance students’ learning ( Cooper et al., 2006 ; Fernández-Alonso et al., 2015 ; Valle et al., 2016 ; Fan et al., 2017 ; Rosário et al., 2018 ). Still, homework is a complex process and needs to be analyzed thoroughly. For instance, when planning and designing homework, teachers need to choose a set of homework characteristics (e.g., frequency, purposes, degree of individualization, see Cooper, 2001 ; Trautwein et al., 2006b ) considering students’ attributes (e.g., Cooper, 2001 ), which may pose a daily challenge even for experienced teachers as those of the current study. Regardless of grade level, quality homework results from the balance of a set of homework characteristics, several of which were addressed by our participants. As our data suggest, teachers need time and space to reflect on their practices and design homework tasks suited for their students. To improve the quality of homework design, school administrators may consider organizing teacher training addressing theoretical models of homework assignment and related research, discussing homework characteristics and their influence on students’ homework behaviors (e.g., amount of homework completed, homework effort), and academic achievement. We believe that this training would increase teachers’ knowledge and self-efficacy beliefs to develop homework practices best suited to their students’ needs, manage work obstacles and, hopefully, assign quality homework.

Ethics Statement

This study was reviewed and approved by the ethics committee of the University of Minho. All research participants provided written informed consent in accordance with the Declaration of Helsinki.

Author Contributions

PR and TN substantially contributed to the conception and the design of the work. TN and JC were responsible for the literature search. JC, TN, AN, and TM were responsible for the acquisition, analysis, and interpretation of data for the work. PR was also in charge of technical guidance. JN made important intellectual contribution in manuscript revision. PR, JC, and TN wrote the manuscript with valuable inputs from the remaining authors. All authors agreed for all aspects of the work and approved the version to be published.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors would like to thank Fuensanta Monroy and Connor Holmes for the English editing of the manuscript.

Funding. This study was conducted at Psychology Research Centre, University of Minho, and supported by the Portuguese Foundation for Science and Technology and the Portuguese Ministry of Education and Science through national funds and when applicable co-financed by FEDER under the PT2020 Partnership Agreement (UID/PSI/01662/2013). PR was supported by the research projects EDU2013-44062-P (MINECO) and EDU2017-82984-P (MEIC). TN was supported by a Ph.D. fellowship (SFRH/BD/80405/2011) from the Portuguese Foundation for Science and Technology (FCT).

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpsyg.2019.00224/full#supplementary-material

Bang H. (2012). Promising homework practices: teachers’ perspectives on making homework work for newcomer immigrant students. High Sch. J. 95 3–31. 10.1353/hsj.2012.0001 [ CrossRef ] [ Google Scholar ]
Bazeley P., Jackson K. (2013). Qualitative Data Analysis with NVivo. London: Sage. [ Google Scholar ]
Bembenutty H. (2011a). Meaningful and maladaptive homework practices: the role of self-efficacy and self-regulation. J. Adv. Acad. 22 448–473. 10.1177/1932202X1102200304 [ CrossRef ] [ Google Scholar ]
Bembenutty H. (2011b). The last word: an interview with Harris Cooper-Research, policies, tips, and current perspectives on homework. J. Adv. Acad. 22 340–350. 10.1177/1932202X1102200207 [ CrossRef ] [ Google Scholar ]
Braun V., Clarke V. (2006). Using thematic analysis in psychology. Qual. Res. Psychol. 3 77–101. 10.1191/1478088706qp063oa [ CrossRef ] [ Google Scholar ]
Brock C. H., Lapp D., Flood J., Fisher D., Han K. T. (2007). Does homework matter? An investigation of teacher perceptions about homework practices for children from nondominant backgrounds. Urban Educ. 42 349–372. 10.1177/0042085907304277 [ CrossRef ] [ Google Scholar ]
Cleary T. J., Chen P. P. (2009). Self-regulation, motivation, and math achievement in middle school: variations across grade level and math context. J. Sch. Psychol. 47 291–314. 10.1016/j.jsp.2009.04.002 [ PubMed ] [ CrossRef ] [ Google Scholar ]
Cooper H. (1989). Synthesis of research on homework. Educ. Leadersh. 47 85–91. [ Google Scholar ]
Cooper H. (2001). The Battle Over Homework: Common Ground for Administrators, Teachers, and Parents , 2nd Edn Thousand Oaks, CA: Sage Publications. [ Google Scholar ]
Cooper H., Robinson J., Patall E. (2006). Does homework improve academic achievement? A synthesis of research. Rev. Educ. Res. 76 1–62. 10.3102/00346543076001001 [ CrossRef ] [ Google Scholar ]
Corno L. (2000). Looking at homework differently. Element. Sch. J. 100 529–548. 10.1086/499654 [ CrossRef ] [ Google Scholar ]
Creswell J. W. (2007). Qualitative Inquiry and Research Method: Choosing Among Five Approaches , 2nd Edn Thousand Oaks, CA: Sage. [ Google Scholar ]
Danielson M., Strom B., Kramer K. (2011). Real homework tasks: a pilot study of types, values, and resource requirements. Educ. Res. Q. 35 17–32. [ Google Scholar ]
Dettmers S., Trautwein U., Lüdtke O. (2009). The relationship between homework time and achievement is not universal: evidence from multilevel analyses in 40 countries. Sch. Effective. Sch. Improve. 20 375–405. 10.1080/09243450902904601 [ CrossRef ] [ Google Scholar ]
Dettmers S., Trautwein U., Lüdtke O., Kunter M., Baumert J. (2010). Homework works if homework quality is high: using multilevel modeling to predict the development of achievement in mathematics. J. Educ. Psychol. 102 467–482. 10.1037/a0018453 [ CrossRef ] [ Google Scholar ]
Epstein J., Van Voorhis F. (2012). “The changing debate: from assigning homework to designing homework,” in Contemporary Debates in Child Development and Education , eds Suggate S., Reese E. (London: Routledge; ), 263–273. [ Google Scholar ]
Epstein J. L., Van Voorhis F. L. (2001). More than ten minutes: teachers’ roles in designing homework. Educ. Psychol. 36 181–193. 10.1207/S15326985EP3603_4 [ CrossRef ] [ Google Scholar ]
Fan H., Xu J., Cai Z., He J., Fan X. (2017). Homework and students’ achievement in math and science: A 30-year meta-analysis, 1986–2015. Educ. Res. Rev. 20 35–54. 10.1016/j.edurev.2016.11.003 [ CrossRef ] [ Google Scholar ]
Fernández-Alonso R., Álvarez-Díaz M., Suárez-Álvarez J., Muñiz J. (2017). Students’ achievement and homework assignment strategies. Front. Psychol. 8 : 286 . 10.3389/fpsyg.2017.00286 [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
Fernández-Alonso R., Suárez-Álvarez J., Muñiz J. (2015). Adolescents’ homework performance in mathematics and science: personal factors and teaching practices. J. Educ. Psychol. 107 1075–1085. 10.1037/edu0000032 [ CrossRef ] [ Google Scholar ]
Foyle H., Lyman L., Tompkins L., Perne S., Foyle D. (1990). Homework and Cooperative Learning: A Classroom Field Experiment. Emporia, KS: Emporia State University. [ Google Scholar ]
Gottfried A. E., Marcoulides G. A., Gottfried A. W., Oliver P. H., Guerin D. W. (2007). Multivariate latent change modeling of developmental decline in academic intrinsic math motivation and achievement: childhood through adolescence. Int. J. Behav. Dev. 31 317–327. 10.1177/0165025407077752 [ CrossRef ] [ Google Scholar ]
Hagger M., Sultan S., Hardcastle S., Chatzisarantis N. (2015). Perceived autonomy support and autonomous motivation toward mathematics activities in educational and out-of-school contexts is related to mathematics homework behavior and attainment. Contemp. Educ. Psychol. 41 111–123. 10.1016/j.cedpsych.2014.12.002 [ CrossRef ] [ Google Scholar ]
Hill C. E., Knox S., Thompson B. J., Williams E. N., Hess S. A., Ladany N. (2005). Consensual qualitative research: an update. J. Couns. Psychol. 52 196–205. 10.1037/a0033361 [ PubMed ] [ CrossRef ] [ Google Scholar ]
Hong E., Wan M., Peng Y. (2011). Discrepancies between students’ and teachers’ perceptions of homework. J. Adv. Acad. 22 280–308. 10.1177/1932202X1102200205 [ CrossRef ] [ Google Scholar ]
Kaur B. (2011). Mathematics homework: a study of three grade eight classrooms in Singapore. Int. J. Sci. Math. Educ. 9 187–206. 10.1007/s10763-010-9237-0 [ CrossRef ] [ Google Scholar ]
Kaur B., Yap S. F., Koay P. L. (2004). The learning of mathematics – expectations, homework and home support. Primary Math. 8 22–27. [ Google Scholar ]
Kitzinger J. (1995). Qualitative research: introducing focus groups. BMJ 311 299–302. 10.1136/bmj.311.7000.299 [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
Krueger R. A., Casey M. A. (2000). Focus Groups: A Practical Guide for Applied Research , 3rd Edn Thousand Oaks, CA: Sage, 10.1037/10518-189 [ CrossRef ] [ Google Scholar ]
Kukliansky I., Shosberger I., Eshach H. (2014). Science teachers’ voice on homework: beliefs, attitudes, and behaviors. Int. J. Sci.Math. Educ. 14 229–250. 10.1007/s10763-014-9555-8 [ CrossRef ] [ Google Scholar ]
Landis J. R., Koch G. G. (1977). The measurement of observer agreement for categorical data. Biometrics 33 159–174. 10.2307/2529310 [ PubMed ] [ CrossRef ] [ Google Scholar ]
Lincoln Y. S., Guba E. G. (1985). Naturalistic Inquiry. Beverly Hills, CA: Sage. [ Google Scholar ]
Margolis H., McCabe P. (2004). Resolving struggling readers’ homework difficulties: a social cognitive perspective. Read. Psychol. 25 225–260. 10.1080/02702710490512064 [ CrossRef ] [ Google Scholar ]
Morgan D. L. (1997). Focus Group as Qualitative Research , 2nd Edn Thousand Oaks, CA: Sage, 10.4135/9781412984287 [ CrossRef ] [ Google Scholar ]
Muhlenbruck L., Cooper H., Nye B., Lindsay J. J. (2000). Homework and achievement: explaining the different strengths of relation at the elementary and secondary school levels. Soc. Psych. Educ. 3 295–317. 10.1023/A:1009680513901 [ CrossRef ] [ Google Scholar ]
National Research Council and Mathematics Learning Study Committee. (2001). Adding it up: Helping Children Learn Mathematics. Washington, DC: National Academies Press. [ Google Scholar ]
Núñez J. C., Rosário P., Vallejo G., González-Pienda J. (2013). A longitudinal assessment of the effectiveness of a school-based mentoring program in middle school. Contemp. Educ. Psychol. 38 11–21. 10.1016/j.cedpsych.2012.10.002 [ CrossRef ] [ Google Scholar ]
Núñez J. C., Suárez N., Cerezo R., González-Pienda J., Rosário P., Mourão R., et al. (2015a). Homework and academic achievement across Spanish compulsory education. Educ. Psychol. 35 726–746. 10.1080/01443410.2013.817537 [ CrossRef ] [ Google Scholar ]
Núñez J. C., Suárez N., Rosário P., Vallejo G., Valle A., Epstein J. L. (2015b). Relationships between parental involvement in homework, student homework behaviors, and academic achievement: differences among elementary, junior high, and high school students. Metacogn. Learn. 10 375–406. 10.1007/s11409-015-9135-5 [ CrossRef ] [ Google Scholar ]
OECD (2014a). PISA 2012 Results in Focus: Does Homework Perpetuate Inequities in Education?, PISA. Paris: OECD Publishing. [ Google Scholar ]
OECD (2014b). PISA 2012 Results in Focus: What 15-Year-Olds Know and what they Can do With What They Know, PISA. Paris: OECD Publishing. [ Google Scholar ]
Patall E. A., Hooper S., Vasquez A. C., Pituch K. A., Steingut R. R. (2018). Science class is too hard: perceived difficulty, disengagement, and the role of teacher autonomy support from a daily diary perspective. Learn. Instr. 58 220–231. 10.1016/j.learninstruc.2018.07.004 [ CrossRef ] [ Google Scholar ]
Peterson E., Irving S. (2008). Secondary school students’ conceptions of assessment and feedback. Learn. Instr. 18 238–250. 10.1016/j.learninstruc.2007.05.001 [ CrossRef ] [ Google Scholar ]
Ramdass D., Zimmerman B. J. (2011). Developing self-regulation skills: the important role of homework. J. Adv. Acad. 22 194–218. 10.1177/1932202X1102200202 [ CrossRef ] [ Google Scholar ]
Richards L. (2005). Handling Qualitative Data: A Practical Guide. London: Sage Publications. [ Google Scholar ]
Rønning M. (2011). Who benefits from homework assignments? Econ. Educ. Rev. 30 55–64. 10.1016/j.econedurev.2010.07.001 [ CrossRef ] [ Google Scholar ]
Rosário P., Cunha J., Nunes A. R., Moreira T., Núñez C., Xu J. (2019). “Did you do your homework?” Mathematics teachers’ perspectives of homework follow-up practices at middle school. Psychol. Sch. 56 92–108. 10.1002/pits.22198 [ CrossRef ] [ Google Scholar ]
Rosário P., Mourão R., Baldaque M., Nunes T., Núñez J., González- Pienda J., et al. (2009). Tareas para casa, autorregulación del aprendizaje y rendimiento en Matemáticas. Revista de Psicodidáctica 14 179–192. [ Google Scholar ]
Rosário P., Mourão R., Trigo L., Suárez N., Fernandéz E., Tuero-Herrero E. (2011). Uso de diarios de tareas para casa en el inglés como lengua extranjera: evaluación de pros y contras en el aprendizaje autorregulado y rendimiento. Psicothema 23 681–687. [ PubMed ] [ Google Scholar ]
Rosário P., Núñez J. C., Vallejo G., Cunha J., Nunes T., Mourão R., et al. (2015). Does homework design matter? The role of homework’s purpose in student mathematics achievement. Contemp. Educ. Psychol. 43 10–24. 10.1016/j.cedpsych.2015.08.001 [ CrossRef ] [ Google Scholar ]
Rosário P., Núñez J. C., Vallejo G., Nunes T., Cunha J., Fuentes S., et al. (2018). Homework purposes, homework behaviors, and academic achievement. Examining the mediating role of students’ perceived homework quality. Contemp. Educ. Psychol. 53 168–180. 10.1016/j.cedpsych.2018.04.001 [ CrossRef ] [ Google Scholar ]
Trautwein U. (2007). The homework-achievement relation reconsidered: differentiating homework time, homework frequency, and homework effort. Learn. Instr. 17 372–388. 10.1016/j.learninstruc.2007.02.009 [ CrossRef ] [ Google Scholar ]
Trautwein U., Köller O. (2003). The relationship between homework and achievement—still much of a mystery. Educ. Psychol. Rev. 15 115–145. 10.1023/A:1023460414243 [ CrossRef ] [ Google Scholar ]
Trautwein U., Köller O., Schmitz B., Baumert J. (2002). Do homework assignments enhance achievement? a multilevel analysis in 7th-grade mathematics. Contemp. Educ. Psychol. 27 26–50. 10.1006/ceps.2001.1084 [ CrossRef ] [ Google Scholar ]
Trautwein U., Lüdtke O. (2007). Students’ self-reported effort and time on homework in six school subjects: between-students differences and within-student variation. J. Educ. Psychol. 99 432–444. 10.1037/0022-0663.99.2.432 [ CrossRef ] [ Google Scholar ]
Trautwein U., Lüdtke O. (2009). Predicting homework motivation and homework effort in six school subjects: the role of person and family characteristics, classroom factors, and school track. Learn. Instr. 19 243–258. 10.1016/j.learninstruc.2008.05.001 [ CrossRef ] [ Google Scholar ]
Trautwein U., Lüdtke O., Kastens C., Köller O. (2006a). Effort on homework in grades 5 through 9: development, motivational antecedents, and the association with effort on classwork. Child Dev. 77 1094–1111. 10.1111/j.1467-8624.2006.00921.x [ PubMed ] [ CrossRef ] [ Google Scholar ]
Trautwein U., Lüdtke O., Schnyder I., Niggli A. (2006b). Predicting homework effort: support for a domain-specific, multilevel homework model. J. Educ. Psychol. 98 438–456. 10.1037/0022-0663.98.2.438 [ CrossRef ] [ Google Scholar ]
Trautwein U., Niggli A., Schnyder I., Lüdke O. (2009a). Between-teacher differences in homework assignments and the development of students’ homework effort, homework emotions, and achievement. J. Educ. Psychol. 101 176–189. 10.1037/0022-0663.101.1.176 [ CrossRef ] [ Google Scholar ]
Trautwein U., Schnyder I., Niggli A., Neumann M., Lüdtke O. (2009b). Chameleon effects in homework research: the homework-achievement association depends on the measures and the level of analysis chosen. Contemp. Educ. Psychol. 34 77–88. 10.1016/j.cedpsych.2008.09.001 [ CrossRef ] [ Google Scholar ]
Valle A., Regueiro B., Núñez J. C., Rodríguez S., Piñeiro I., Rosário P. (2016). Academic goals, student homework engagement, and academic achievement in elementary school. Front. Psychol. 7 : 463 . 10.3389/fpsyg.2016.00463 [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
Wang M.-T., Eccles J. S. (2012). Adolescent behavioral, emotional, and cognitive engagement trajectories in school and their differential relations to educational success. J. Res. Adolesc. 22 31–39. 10.1111/j.1532-7795.2011.00753.x [ CrossRef ] [ Google Scholar ]
Xu J. (2010). Homework purposes reported by secondary school students: a multilevel analysis. J. Educ. Res. 103 171–182. 10.1080/00220670903382939 [ CrossRef ] [ Google Scholar ]
Xu J. (2015). Investigating factors that influence conventional distraction and tech-related distraction in math homework. Comput. Educ. 81 304–314. 10.1016/j.compedu.2014.10.024 [ CrossRef ] [ Google Scholar ]
Xu J. (2018). Reciprocal effects of homework self-concept, interest, effort, and math achievement. Contemp. Educ. Psychol. 55 42–52. 10.1016/j.cedpsych.2018.09.002 [ CrossRef ] [ Google Scholar ]
Xu J., Yuan R. (2003). Doing homework: listening to students’, parents’, and teachers’ voices in one urban middle school community. Sch. Commun. J. 13 23–44. [ Google Scholar ]
Zakharov A., Carnoy M., Loyalka P. (2014). Which teaching practices improve student performance on high stakes exams? Evidence from Russia. Int. J. Educ. Dev. 36 13–21. 10.1016/j.ijedudev.2014.01.003 [ CrossRef ] [ Google Scholar ]

ORIGINAL RESEARCH article

“homework feedback is…”: elementary and middle school teachers’ conceptions of homework feedback.

$\r\nJennifer Cunha$

1 Departamento de Psicologia Aplicada, Escola de Psicologia, Universidade do Minho, Braga, Portugal
2 Departamento de Psicología, Universidad de Oviedo, Oviedo, Spain

This study explored mathematics teachers’ conceptions of the homework feedback focusing on four key aspects: definition, purpose, types, and perceived impact. Forty-seven teachers from elementary and middle schools participated in six focus groups. Data were analyzed using content analysis. To enhance the trustworthiness of findings, classroom observations were used for triangulation of data. Participants conceptualized homework feedback in three directions (i.e., teachers’ feedback provided to students, students’ feedback provided to teachers, and homework self-feedback), being teachers’ monitoring of students’ learning the purpose reported by most teachers. Participants also reported the types of homework feedback more frequently used in class (e.g., checking homework completion, checking homework on the board), and their perceived impact on students. Findings provide valuable information to deepen the understanding of the homework feedback process, which may help develop new avenues for future research.

Introduction

Homework may be defined as tasks assigned by teachers to be completed in non-instructive time ( Cooper, 2001 ), and has proved to enhance students’ academic achievement when endowed with particular characteristics (e.g., short, purposeful, frequent assignments, high quality) (e.g., Dettmers et al., 2010 ; Fernández-Alonso et al., 2015 ; Rosário et al., 2015a ).

In addition, the homework feedback provided by the teacher in class is an important tool to increase the impact of homework on students’ learning and academic achievement (e.g., Walberg and Paik, 2000 ; Núñez et al., 2015b ; Rosário et al., 2015b ), and a crucial aspect of the quality of homework ( Cooper, 2001 ). However, detailed information on elementary and middle school teachers’ perspectives about their practices and on the reasons why teachers choose and use particular types of homework feedback in class is still scarce ( Bang et al., 2009 ; Deslandes, 2009 ; Rosário et al., 2015b ). Investigating teachers’ conceptions of the homework feedback, particularly in elementary and middle school, may provide new insights into research on homework (e.g., helping further explain previous quantitative results; improving homework feedback measures), as well as into educational practices (e.g., teachers getting training on homework feedback practices).

Teachers’ Role on the Homework Feedback Process

Teachers play an important role in the first phase of the homework process by setting up the objectives of homework assignments and designing tasks, and also in the final phase by implementing classroom follow-up practices ( Cooper, 2001 ; Epstein and Van Voorhis, 2001 ; Rosário et al., 2015a , b ). The latter includes, among other practices, homework feedback provided in class: oral or written praise, criticism, written comments (highlighting right and wrong answers), rewards, general review of homework in class, and grading (i.e., teachers giving numerical grades) (e.g., Elawar and Corno, 1985 ; Murphy et al., 1987 ; Corno, 2000 ; Cooper, 2001 ). These homework feedback practices are an important instructional tool for teachers in their teaching process (e.g., helping identify students’ difficulties, errors or misconceptions in homework; approaching the learning contents to accommodate students’ lack of prior knowledge, and redesigning homework to match students’ needs) ( Corno, 2000 ; Walberg and Paik, 2000 ; Epstein and Van Voorhis, 2001 ; An and Wu, 2012 ; Bang, 2012 ).

Extant research lacks studies which have focused specifically on each of the above-mentioned types of homework feedback; still, some studies have shed some light on the usage and benefits of the various types of homework feedback. For example, Murphy et al. (1987 , p. 68) found that “class discussion on homework,” and grading and commenting on homework were the practices most frequently used by high school teachers (i.e., English, mathematics, science, and social science) to monitor students’ completion of homework. Focusing on mathematics, Kaur (2011) explored the nature of homework tasks assigned by three 8th grade mathematics teachers (e.g., types of homework, sources of homework tasks), and found that teachers provided feedback on errors by grading assignments, orienting discussions and checking homework on the board when needed. Using the TIMSS 2003 data set, Zhu and Leung (2012) found that a high percentage of 8th grade mathematics teachers reported checking homework completion (85%), providing feedback regularly (i.e., at least “sometimes”, 100%), and discussing homework in class (96%). Nevertheless, none of these studies deeply explored the process of homework feedback.

Students’ Role on the Homework Feedback Process

Students engaging in school tasks with autonomy and responsibility are expected to develop a sense of personal agency for self-managing their behaviors ( Zimmerman, 1989 ). Besides, students who proactively manage their behaviors to attain self-set goals are likely to self-regulate their learning efficiently ( Zimmerman, 1989 ). From a social cognitive perspective, self-regulated learning (SRL) may be defined as an active learning process whereby students self-set goals that direct their cognitions, motivations and behaviors toward those goals ( Zimmerman, 1989 ; Núñez et al., 2013 ; Rosário et al., 2013 ). For example, robust self-efficacy and autonomy, good study skills, commitment to self-set goals, and positive academic attitudes are examples of core elements of academic self-regulation which are necessary to complete homework ( Zimmerman and Kitsantas, 2005 ; Ramdass and Zimmerman, 2011 ; Schmitz and Perels, 2011 ; Núñez et al., 2015c ). Regarding the latter, extant literature highlighted self-regulation competencies as essential tools not only to help students complete their homework, but also use the homework feedback delivered with efficacy and responsibility ( Ramdass and Zimmerman, 2011 ; Zhu and Leung, 2012 ; Xu and Wu, 2013 ). In fact, students are given homework feedback in class and play an important role deciding what to do next with the information given (e.g., ignoring feedback, self-evaluating their homework performance, using SRL learning strategies).

However, to authors’ knowledge, research has not yet provided information to contribute to understanding how teachers’ homework feedback may promote students’ active role in the homework feedback process. As Corno (2000) reported, teachers are expected to promote students’ capacity to self-evaluate their homework, which would involve addressing important self-regulatory processes. Otherwise, homework feedback may fail to benefit students ( Zhu and Leung, 2012 ).

The Benefits of Homework Feedback

Research has analyzed the effect of specific types of homework feedback provided by teachers on students’ academic performance in a particular subject (e.g., Elawar and Corno, 1985 ; Rosário et al., 2015b ), and also the relationships between homework variables (e.g., homework feedback perceived by students, students’ interest, homework management) using non-subject-centered designs ( Xu, 2012 ; Núñez et al., 2015b ). Focusing on the former (i.e., investigating homework feedback in a particular subject), Cardelle and Corno (1981) examined the effect of three types of written homework feedback (i.e., praise, constructive criticism, and constructive criticism plus praise) on college students’ written performance in a second language. Findings showed that students under the constructive criticism plus praise condition achieved a better written performance than their counterparts. Moreover, irrespective of performance levels (i.e., high, middle, and low), participants reported their preference for the constructive criticism plus praise condition. Elawar and Corno (1985) conducted a similar study in mathematics with 6th grade students. Findings showed that students under the constructive criticism plus praise condition showed better achievement and a more positive attitude toward mathematics (e.g., enthusiasm for mathematics) than students of the control group.

The synthesis by Walberg et al. (1985) , and also recent findings by Rosário et al. (2015b) , indicated that specific and individual feedback (i.e., giving written comments or grading homework) positively impacts students’ academic achievement. However, checking homework, grading, and providing individual feedback on homework assignments for every single student in class may not always be feasible because of teacher’s heavy workload (e.g., large numbers of students per class, large numbers of classes to teach, many school meetings per week) ( Cooper, 2001 ; Rosário et al., 2015b ). This educational constraint may help explain why checking homework on the board and checking homework orally are among the homework follow-up practices most frequently used by English as a Foreign Language (EFL) teachers ( Rosário et al., 2015b ). These practices are useful to teachers because they allow providing feedback to the whole class (e.g., Brookhart, 2008 ) with less effort than that needed to grade homework or comment on students’ assignments.

Moreover, homework feedback perceived by students was also investigated using non-subject-centered designs (e.g., Xu, 2011 ; Núñez et al., 2015b ; for exceptions see Tas et al., 2016 ; Xu et al., 2017 ). In general, findings showed some of the benefits of homework feedback for students. For example, Xu’s studies using multilevel designs found that at student level teachers’ homework feedback reported by 8th and 11th grade students was positively associated with students’ interest in homework ( Xu, 2008 ), students’ reasons for doing homework ( Xu, 2010 ), students’ homework management ( Xu and Wu, 2013 ; Xu et al., 2017 ), and students’ homework motivation management ( Xu, 2014 ).

Analyzing students’ homework completion at 8th and 11th grade levels, Xu (2011) found a positive association with teacher homework feedback at both student level and class level. The explained variance was higher at class level. The author concluded that students’ homework completion is related to teachers’ provision of homework feedback ( Xu, 2011 ). This proposition is further substantiated by the findings by Bang (2011) , showing that high school immigrant students perceived teachers’ feedback as a facilitating factor, and the lack of it as an obstacle to homework completion.

More recently, Núñez et al. (2015b) conducted a study with students from various school years (grades 5–12) and concluded that the stronger the teachers’ homework feedback is perceived by students, the greater the amount of homework completed and the better the quality of homework time management (e.g., how well students managed time devoted to homework and avoided distractions). Moreover, these authors found that students’ academic achievement is indirectly and positively associated with homework feedback through students’ homework behaviors (i.e., amount of homework completed) and self-regulation (i.e., quality of homework time management), highlighting the importance of student engagement in the homework process ( Núñez et al., 2015b ). The results of Tas et al. (2016) are consistent with those, showing that middle school students’ homework self-regulation (e.g., orientation goals, learning strategies) mediated the relationship between perceived homework feedback and science achievement.

Bang (2012) reported that teachers acknowledged homework feedback (i.e., grading homework) as an important tool to motivate immigrant students to complete homework. Still, teachers admitted the educational challenge of providing homework feedback because of the time-consuming nature of this strategy. In fact, Rosário et al. (2015b) also reported the difficulties faced by EFL teachers to collect and grade homework on a regular basis. Both studies ( Bang, 2012 ; Rosário et al., 2015b ) called for further research on teachers’ perspectives about homework feedback.

In spite of the benefits of homework feedback for students previously reported, the literature has shown that teachers’ support in homework perceived by students decreases from elementary to middle school ( Katz et al., 2010 ; Núñez et al., 2015b ), without specifying in what aspects. Moreover, Kukliansky et al. (2016) recently observed middle school teachers’ behaviors in science classes (3–5 consecutive times) and found that in-class instructional feedback was not always provided, even when demanded by students. However, the authors did not explore the reasons why teachers did not provide feedback in this situation.

In sum, extant research on homework feedback has been conducted on controlled domain-centered contexts (e.g., Elawar and Corno, 1985 ; Rosário et al., 2015b ), on single grade levels (e.g., Kaur, 2011 ; Zhu and Leung, 2012 ; Xu et al., 2017 ), is non-subject-centered (e.g., Xu, 2011 ), or explored specific populations (e.g., teachers of immigrant students, Bang, 2012 ) (cf. Table 1 ), thus further research is needed to deepen the understanding of the homework feedback process.

TABLE 1. Summary of studies that focus homework feedback.

The Present Study

Teachers are an important source of information in the study of the homework feedback process because they actually manage feedback in class ( Cooper, 2001 ). Still, little is known about how mathematics teachers of different school levels perceive homework feedback. Examining elementary and middle school teachers’ conceptions of the homework feedback is expected to reveal useful information on the homework process, especially teachers’ beliefs and practices concerning homework feedback (cf. Irving et al., 2011 ). The model of teachers’ conceptions of assessment and feedback ( Irving et al., 2011 ; see also Peterson and Irving, 2008 ) provides a relevant theoretical framework for the current research, and guided the research questions, data collection and analysis. This model addresses four key aspects of assessment and feedback: definition, purpose, personal response (i.e., types of assessment and feedback used) and perceived impact. Analyzing these key aspects focused on the homework feedback may provide data to help explain previous findings showing small effect sizes or low explained variances (see Zhu and Leung, 2012 ; Xu, 2014 ; Núñez et al., 2015b ; Rosário et al., 2015b ), and design future studies, homework policies or school-based interventions.

The following research questions guided the current study:

What are elementary and middle school teachers’ conceptions of homework feedback?

How do the four key aspects of the homework feedback relate to each other?

The current study explores the conceptions of teachers of two school levels for two reasons. Firstly, there are some differences as to the educational goals of those school levels; while teachers at elementary school focus on working on the foundations of mathematics (e.g., giving support in the development of number sense), middle school students are expected to learn high-level concepts (e.g., application of proportional relationships). Secondly, homework research found that the characteristics of the homework assigned (e.g., amount of homework assigned, homework purposes) vary for elementary and middle school. For example, Mullis et al. (2004) found that middle school students are expected to do larger amounts of homework than elementary school students. Besides, the purposes of assignments may also vary for both school grades. While homework purposes for middle school may be more related to school contents assessed in tests, homework purposes for elementary school are more likely to aim at developing personal skills such as time management (e.g., Muhlenbruck et al., 1999 ; Cooper and Valentine, 2001 ). Notwithstanding, the recent meta-analysis focused on mathematics and science by Fan et al. (2017) included a study in which elementary school teachers reported to assign homework to practice basic mathematics skills (see Bedford, 2014 ). Those differences (e.g., amount of homework assigned, homework purposes) may help explain the differential results regarding the benefits of homework in elementary and middle school (e.g., Cooper et al., 2006 ; Fan et al., 2017 ). Hence, elementary and middle school teachers were invited to talk about homework feedback in order to learn their conceptions and reported practices.

The current study focuses on mathematics (see Trautwein et al., 2006 on the importance of focusing homework research on a specific domain). The reason is threefold: students’ achievement levels, educational relevance of the subject, and previous research findings on homework. There is a global educational concern about students’ poor performance in mathematics. The PISA 2012 report indicates that students from 35 countries show a mathematics performance below the OECD average ( OECD, 2014 ). This worrying educational scenario raises serious challenges for some countries (among which is Portugal), given the fundamental role played by mathematics in other subjects (e.g., biology, physics) and in the development of life and citizenship skills (e.g., Reyna and Brainerd, 2007 ; OECD, 2014 ; Hagger et al., 2015 ). Moreover, mathematics was chosen because of the great amount of homework that is regularly set by teachers (e.g., Trautwein et al., 2006 ; Schmitz and Perels, 2011 ; Xu, 2015 ).

Materials and Methods

School and participants characteristics.

The last 2 years of elementary school in the Portuguese educational system encompass 5th and 6th grades (10 and 11 years old), while middle school includes 7th, 8th, and 9th grade (12–14 years old). Students have 270 min of mathematics per week in 5th and 6th grade, and 225 min per week in each of the three middle school years. At the end of 6th and 9th grade students complete a final exam that counts toward 30% of the overall grade.

Homework is an educational tool often used by Portuguese teachers as part of their lessons; still, there are no formal homework policies for Portuguese public schools (e.g., characteristics of homework assignments, homework follow-up practices; Rosário et al., 2015b ).

In the current study, participants were involved in focus group discussions and some of them in classroom observations.

Participants in Focus Groups

Six focus group discussions were conducted in this study, each of which comprised 7–9 mathematics teachers. Three focus groups were set up with elementary school teachers (5th and 6th grade) and three focus groups with middle school teachers (7th, 8th, and 9th grade). Following Morgan (1997) , homogeneity of groups was ensured in order to encourage participation among participants and minimize inhibition. Participating teachers met the following criteria: (i) having experience in teaching mathematics at elementary or middle school for at least 2 years, and (ii) assigning homework and providing homework feedback regularly (at least once a week). These requirements aimed to guarantee participants’ ability to generate ideas and opinions to share in their focus group.

The school administrators from the pool of schools which had previously enrolled in other university research projects were contacted by the authors. From those schools who agreed to participate, 20 public schools (approximately 25%) were randomly selected, and 75 mathematics teachers (approximately 25% of the pool of available elementary and middle school teachers) were randomly selected. Teachers were e-mailed about the purposes and procedures of the study (e.g., duration of the session, videotaping of the session) and invited to participate. To encourage participation (see Krueger and Casey, 2010 ), teachers were offered a participation reward (i.e., gift card), free baby-sitting services and a 3-h seminar on homework process and SRL to be held after the study had concluded.

In the end, 47 mathematics teachers (an acceptance rate of 63%) from 12 schools agreed to participate in the present study. The first author phoned the volunteer teachers to schedule the focus group meeting. Then, teachers were distributed into the various groups considering criteria such as: school, school level, and preferred scheduled time. Teachers with a hierarchical relationship were not allocated in the same focus group because this might affect their responses and the dynamics of the discussion ( Kitzinger, 1995 ; Irving et al., 2011 ). In order to encourage attendance, all participants were reminded of the focus group session 1 week before and were asked to arrive 10 min early. A map with the location was sent to all participants.

All teachers attended the focus group discussions on the scheduled day (see Table 2 for focus group demographics). Twenty-four teachers (51.1%) were teaching at elementary school level, and 23 (48.9%) at middle school level. In general, participating teachers had 21 years of teaching experience ( SD = 6.11); taught students from middle-class families, as evidenced by the low percentage of students receiving free or reduced-price lunch (19.7%, data collected from the secretary’s office of the participating schools).

TABLE 2. Summary of demographic information of the focus groups.

Participants in Classroom Observations

Given the time-consuming nature of observational studies, of these 47 teachers, 25% of the participants were randomly selected and asked to be observed in their mathematics classes. Finally, six teachers of each school level ( N = 12; four males) were observed in their classrooms. These teachers had been teaching between two to five classes and they had an average of 19 years of teaching experience ( SD = 6.93).

Data Collection

Data was collected from two data sources: focus groups and classroom observations. The research team had previously enrolled in a qualitative research course offered by the University of Minho. Following a hands-on approach, the course training addressed topics including the following: how to lead focus group discussions (e.g., encouraging participation) and observations, and how to ensure the quality and credibility of a qualitative study.

This study was carried out in accordance with the recommendations of the ethics committee of the University of Minho. All subjects gave written informed consent to the different phases of the research (i.e., focus groups and classroom observations) in accordance with the Declaration of Helsinki.

Focus Group Discussions

Focus group interviews allow for in-depth exploration of meanings, attitudes, and personal experiences of participants about a particular topic during an informal, but structured, group discussion ( Kitzinger, 1995 ; Krueger and Casey, 2010 ). This method of data collection helps capture teachers’ tacit knowledge in order to fill research gaps ( Ryan et al., 2014 ). The focus group interviews were conducted by two members of the research team as facilitators while a third member filmed the sessions. To meet teachers’ availability requirements to participate, four focus group discussions were held at the end of the school year (July), and two at the beginning of the following school year (October). Each focus group session lasted approximately 60 min and took place in a room with appropriate light and sound conditions. To create a friendly environment, snacks and refreshments were offered to participants before and after the discussion. The chairs were arranged in a half circle to allow participants to see each other and to facilitate the filming of everyone in the room.

Prior to the discussions, teachers filled in a socio-demographic questionnaire (e.g., gender, years of teaching experience) and signed the written informed consent form. Then, the facilitators introduced themselves, read aloud the study purpose and the basic rules of the focus group discussion, and ensured confidentiality of participants’ responses (i.e., any information that may identify participants or their schools was eliminated at the end of the study).

To facilitate the interaction between participants, all focus group sessions started with a warm-up activity. Then, the facilitators started the discussion with general questions (e.g., the importance of homework) and, following Peterson and Irving (2008) and Irving et al. (2011) , specific questions related to the four key aspects of homework feedback were asked: definition, purpose, types of homework feedback, and perceived impact (see Table 3 ). This set of questions was previously asked to two teachers in order to ensure comprehensibility. These teachers did not participate in the focus group discussions.

TABLE 3. Key areas and guiding questions used in teachers’ focus groups.

Classroom Observations

Classroom observations were conducted to capture teachers’ spontaneous behaviors regarding the homework feedback process. All invited teachers were informed that they would be observed five times on average (see Kukliansky et al., 2016 ), in a period of 3 weeks in the middle of the school year (March). Teachers were blind to the exact date or timetable of the observations (dates of the mathematics assessment tests were excluded from the observations schedule) and all agreed to participate acknowledging these requirements. Two other members of the research team, who were knowledgeable about homework research, conducted the classroom observations. These observations incorporated a structured content based on previous homework research to direct researchers’ attention to teacher’s responses to students’ homework completion (see Choo et al., 2015 ). The instrument used to collect data included the five homework feedback types reported in the literature (e.g., Rosário et al., 2015b ). Additionally, researchers took field notes independently on the homework feedback process (e.g., time spent and how homework feedback was delivered), cross-checked and expanded upon their notes as promptly as possible. In the end, each teacher was observed on average five times, thus gathering a total of 64 h of classroom observations.

Data Analysis

Transcriptions of focus group discussions and observation field notes were analyzed using content analysis ( Bardin, 1996 ). The latter is a qualitative research technique used to search for and identify categories, following systematic procedures ( Bardin, 1996 ).

The researchers who conducted the focus groups carried out the data analysis. Content analysis followed three main steps ( Bardin, 1996 ): (i) reading the focus groups’ verbatim transcriptions to get an overview of the data (pre-analysis), (ii) coding (exploration of data), and (iii) treatment (e.g., percentages) and interpretation of data (e.g., comparing frequencies of coded categories). The organization, management, coding, and querying process of the data were conducted using the QSR International’s NVivo 10 software (e.g., Richards, 2005 ).

The extensiveness of comments (i.e., number of participants who convey an idea, Krueger and Casey, 2000 ) in the current study was the criterion used to identify categories. The identification of categories followed a deductive and inductive iterative process ( Bardin, 1996 ). The categories were organized a priori in a coding scheme based on the theoretical model by Irving et al. (2011) , and on the homework research (e.g., Walberg et al., 1985 ; Cooper, 2001 ; Xu, 2011 ; Rosário et al., 2015b ). For example, the categories “definition,” “purposes,” “types,” and “perceived impact” of homework feedback were driven by the Irving et al. (2011) theoretical model, while each type of homework feedback (e.g., subcategory “checking homework on the board”) was driven by homework research (e.g., Rosário et al., 2015b ). New categories were added during the analysis using participants’ words ( Bardin, 1996 ). For example, the subcategories “homework feedback provided to teacher,” “self-esteem,” “homework self-feedback” were subcategories build upon teachers’ words. In the end, all transcripts were reviewed in order to check whether the already coded material fit the new subcategories.

Finally, the two researchers reviewed all the categories and sub-categories and discussed the differences found in order to reach a consensus (e.g., elimination of the subcategory “teachers assess students’ progress” because it was highly related to the subcategory “teachers monitor students’ learning”). After the data analysis of four focus group discussions (two from each school level), the researchers coded the two other focus group discussions and no new information was added. To ensure the reliability of findings, the Kappa value was calculated using the Coding Comparison Queries in the Navigation View of the NVivo software. The Kappa value was 0.86, which may be considered “almost perfect” according to Landis and Koch (1977 , p. 165). Then data from the elementary and middle school teachers were analyzed separately conducting a matrix-coding query, crossing nodes with attributes (i.e., school level). The number of participants for each subcategory was converted into a percentage.

The two researchers who conducted the classroom observations coded independently the process of homework feedback delivery described in the field notes according to the codebook used in the focus groups. No new categories or subcategories were identified or redefined. Data from the elementary and middle school teachers were analyzed separately following the procedure used in focus groups, and the number of participants for each subcategory was converted into a percentage. To avoid bias on the Kappa value in NVivo, due to different numbers of characters of the researchers’ field notes ( Kim et al., 2016 ), data was exported and IBM SPSS was used to calculate Cohen’s Kappa for nominal variables. The Cohen’s Kappa value for each subcategory ranged between 0.81 and 1.0, which indicates high agreement across observers.

To answer the second research question (i.e., How do the four key aspects of the homework feedback relate to each other?), data analysis followed two steps using the same software. First, a Cluster Analysis Wizard by word similarity between nodes was conducted to explore patterns and connections between nodes in an initial phase of data analysis ( Bazeley and Jackson, 2013 ). Second, a case-by-nodes matrix was conducted to explore the relationships between each category in the focus group discussion transcripts as suggested by Bazeley and Jackson (2013) .

Specific quality procedures were used to enhance the trustworthiness of the findings of the current study ( Lincoln and Guba, 1985 ): investigator triangulation (i.e., several investigators were involved in the analysis process), methodological triangulation (i.e., patterns in data from focus groups and classroom observations were compared using a matrix-coding query, crossing nodes with classified sources – focus group and observations), and a member checking run at the University facilities. The researchers randomly selected and invited 25% of the participants of each grade level to do a member check ( Lincoln and Guba, 1985 ). Ten teachers agreed to participate (six from elementary school and four from middle school) (response rate of 83%). Member checking session lasted approximately 2 h. Firstly, participants were informed of the findings (approximately 45 min). Afterward, they were given a copy of the findings and asked to analyze and discuss whether the description was an authentic representation of the topics covered during the focus group interviews. The participants also analyzed whether the description of the homework feedback types provided to students was an authentic representation of what usually happens in class. Participants were invited to critically analyze the findings and comment on them ( Creswell, 2007 ).

Data were organized and reported according to each of the key aspects of teachers’ conceptions of the homework feedback (see Peterson and Irving, 2008 ): definition, purpose, types of homework feedback practices, and perceived impact of homework feedback (see Figures 1 , 2 ). Furthermore, the relationships between these four aspects were presented (see Figure 3 ). Teachers’ verbatim quotes were introduced to illustrate the categories and conversations held in the focus group discussions (see also Table 4 ). In addition, whenever possible, data from classroom observations were included to illustrate findings. Categories were reported using the criteria by Hill et al. (2005 , p. 16) as follows: general (i.e., categories include all, or all but one, of the cases), typical (i.e., categories include more than half of the cases) and variant (i.e., categories include more than three cases or up to half of the cases). For reasons of parsimony, rare categories (two or three cases) were not reported.

FIGURE 1. Elementary and middle school teachers’ conceptions of homework feedback for each key aspect.

FIGURE 2. Observed elementary and middle school teachers’ homework feedback practices.

FIGURE 3. Relationships among teachers’ conceptions of homework feedback.

TABLE 4. Summary of findings.

Initial Data Screening

All participants reported assigning homework regularly and considered homework feedback as an important element for homework effectiveness. Data showed that, for each homework assignment, 96% of the elementary school teachers and 52% of the middle school teachers reported spending approximately 30 min giving homework feedback in class. Moreover, 48% of the middle school teachers spent on average 15 min giving homework feedback in class. Classroom observations provided precise information on the time spent in class giving homework feedback: 3–80 min in elementary school classes ( M = 32.75; SD = 19.91), and 5–55 min in middle school classes ( M = 29.89; SD = 17.36).

Definitions of Homework Feedback

When teachers were asked about their definition of homework feedback, the majority said they “had never thought about it” (F1P1). Still, elementary and middle school teachers elaborated on homework feedback differently (see Figure 1 ). Teachers from elementary school proposed two meanings for homework feedback: (i) homework feedback provided by the teacher and (ii) students’ homework self-feedback. For middle school teachers, homework feedback was conceptualized as threefold): (i) homework feedback provided by the teacher; (ii) homework feedback provided by the student; and (iii) students’ homework self-feedback. The analysis of the frequency labels for each category revealed no general categories, which allows concluding that definitions of homework feedback vary among teachers, irrespective of the grade level. Moreover, while “homework feedback provided to students” is a variant category in elementary school, in middle school is a typical category (see Figure 1 ).

For elementary school teachers in one focus group discussion and for middle school teachers in two focus groups, homework feedback provided by teachers was defined as a message provided to students with information concerning their homework behaviors (i.e., completion, effort), and comprehension of homework tasks and performance (e.g., how well students answered, why answers are wrong).

Middle school teachers in all focus groups conceptualized homework feedback in the reverse direction (i.e., from the students to the teacher), as the following statement illustrates:

F5P2: Some weeks ago, I noticed that several students in class had not understood some homework exercises. I asked the whole class and found out that no one had understood. Two or three students said: Sir, these exercises were a bit complicated… We did not understand what we were expected to do, how to start… This was the homework feedback they gave me.

The remaining teachers nodded their heads in agreement and added that this piece of information gathered at the beginning of a lesson helps them choose the type of homework feedback to give to students.

Lastly, elementary school teachers of two focus groups, and middle school teachers of one focus group (see Figure 1 ) proposed another meaning for homework feedback: “homework self-feedback” (typical category in elementary school and variant category in middle school). The following utterance illustrates this conceptualization:

F2P3: Homework feedback is when students can explain or reflect upon what they are doing…or checking from their seats when we check homework on the board.

Another elementary school teacher elaborated on students’ homework self-feedback:

F4P4: Homework feedback is also related to students’ homework completion. All my students draw a grid in their notebooks and devote one row to homework. Every day they write 1 for “completed” homework and 0 for “missing.” At the end of the term they have a score. I believe this to be self-feedback because students know their score and link it to school grades. They know that those who complete homework are likely to achieve better results. The opposite is also true….

This type of homework feedback (i.e., self-feedback) is more focused on students’ homework behaviors than on students’ homework performance. Still, other teachers from the same focus group reported that they do not use this strategy with their students.

Purposes of Homework Feedback

The homework feedback purposes identified by teachers at both school levels were similar. Teachers enthusiastically talked about homework feedback as a “working tool” serving three main purposes (see Figure 1 ): (i) teachers monitor students’ learning and behavior (typical category in both school levels); (ii) students monitor their own learning (typical category in elementary school and variant category in middle school); and (iii) promotion of students’ self-esteem (variant category in both school levels).

When asked to expand on this idea, participants explained that homework feedback helps teachers identify students’ difficulties and monitor their content knowledge, which provides information to self-evaluate the instruction process and introduce changes if necessary. In fact, some students struggle to learn and show difficulties to understand and complete homework. To promote students’ motivation to do homework, teachers agreed on the need – “after charging our batteries of patience” (F6P3) – to explain in class how to do homework exercises. Besides, teachers exemplified the usefulness of homework feedback for monitoring students’ homework behaviors (e.g., checking whether students have completed their homework, whether they have copied the solutions from a textbook). This category emerged in all focus group discussions, and was consensual among participants. Teachers emphatically agreed on the examples discussed and expanded on others’ ideas. The following statements illustrate some of the conversations held:

F2P4: Homework feedback is important in order to learn about what is happening on earth [some teachers laughed], to learn whether most students do their homework, whether they manage to do it alone or need some help, but also to learn about their difficulties during the learning process and act upon their mistakes.

F2P1: To know whether I delivered the message well or not so… Homework feedback should make us change our instruction methodologies. If the message was not properly delivered, it’s necessary to change the course of action…

Moreover, many elementary and some middle school teachers in all focus groups mentioned students’ monitoring of their own learning as an important purpose of homework feedback, as illustrated by the following opinion:

F5P3: With the help of homework feedback, students can learn what is right or wrong in their homework. If the homework assignment is correct, they get some positive reinforcement. If it is not correct, they learn that they have to study more and do additional exercises.

Some points made by participants focused exclusively on one of the two previous purposes (see Figure 1 ). However, some teachers in all focus groups irrespective of grade level considered homework feedback a purposeful tool for teachers or students to monitor progress in learning. In sum, teachers admitted that homework feedback provides on-task opportunities for teachers and students to monitor the teaching and learning process.

Moreover, teachers from five focus groups pointed at the promotion of students’ self-esteem as another purpose of homework feedback (see Figure 1 ). Elementary and middle school teachers supported this idea, showing concern about students’ wellbeing, mainly of low achievers:

F5P6: When they [students] realize that they are capable of doing homework exercises, they feel very happy and proud of themselves. When they fail to complete or feel frustrated because they couldn’t find a way to do the exercises, I try to make positive comments, highlighting what they did well in order to make them feel confident. It is crucial to give them positive reinforcements to improve their self-esteem.

Types of Homework Feedback Practices

Going further in the discussion, teachers identified the most frequently used homework feedback practices: (i) checking homework completion; (ii) checking homework on the board; (iii) testing related content; (iv) considering homework in the overall grade; (v) informing parents of their children’s homework non-compliance (homework feedback to parents); and (vi) giving written comments (see Figure 1 ).

The two types of homework feedback practices first mentioned in all focus group discussions were: checking who completes homework and checking homework on the board. The classroom observations (see Figure 2 ) provided information on the classroom routines and, with some exceptions, allowed concluding that classes usually begin with similar routines: checking who did homework and then checking homework on the board.

As Figure 1 shows, checking homework completion is a general and typical category among elementary and middle school teachers, respectively. When discussing this practice (see Table 4 ), some of the elementary and middle school teachers argued that they simply ask who completed homework. On the other hand, most elementary and some middle school teachers explained that they walk around the class having a glance at students’ notebooks in order to check homework completion. This strategy allows noting who actually did their homework and gathers information on how students did it (e.g., whether students followed all the steps to solve a problem). In this process, teachers reported that they try to understand the reasons why students did not complete homework (e.g., is failing to complete homework a class problem or is it only associated with a particular student?). The participating teachers considered this type of homework feedback useful because it gives information on the process and allows them to respond to students’ maladaptive homework behaviors (e.g., missing homework, copying solutions from peer students, writing down results without checking). Teachers from both school levels reported using logs in class to record who missed homework, and data from the classroom observations corroborated this finding. When asked how they usually deal with maladaptive homework completion behaviors, some teachers at both school levels reported criticizing students who repeatedly fail to complete homework or copy answers from the textbook, as the following utterance illustrates:

F6P5: Where is your homework? Oh, I see. Keep working like this and you will get far… [ironic tone]

The use of public criticism and irony in response to maladaptive homework behaviors was observed sometimes in elementary school classes, and often in middle school classes.

When discussing the best practice regarding homework, participants at both school levels named checking homework on the board as a practice that “reaches all students” (F6P4). All teachers were very emphatic about the importance and usefulness of this type of feedback. As Figure 1 shows, this practice is the most frequently used by elementary and middle school teachers. Moreover, present data (i.e., focus group discussions and classroom observations) suggest several ways in which this type of homework feedback may be put into practice. For example, some teachers reported that they check homework on the board; others mentioned writing on the board the answers dictated by students from their seats; while others explained that they randomly choose one or more students to do homework exercises on the board. Elementary and middle school teachers further explained additional homework feedback practices adopted after displaying the solution for the exercise on the board: (i) whole-class discussion led by the teacher; (ii) further explanation provided by the teacher or by the students on what is written on the board; (iii) teachers’ praise for students’ efforts in learning or good performance, or (iv) general incentives encouraging students to persist when doing homework. The observations conducted in the classrooms provided data that showed that all these strategies were used in class when teachers were checking homework on the board. Still, frequency and sequence of the strategies used by teachers (e.g., students check homework on the board, teacher explains problem solving procedures, class discussion) varied according to the needs and characteristics of the class. Moreover, classroom observations revealed that when students ask teachers for help, some teachers provide individual explanations while checking homework on the board. For example, when students raise their hand to show a lack of understanding while checking homework on the board, some teachers go to the student’s desk to answer their question individually.

Teachers at both school levels also emphasized checking homework on the board as a way of giving feedback to the whole class with minimum time and effort:

F1P1: When homework is being checked on the board by a student, I identify what is incorrect and explain how the exercise may be approached. Still, this feedback is very general because I cannot check every single assignment that students hands in. I simple cannot do it!

However, some participating teachers alerted that students who check homework on the board get a more detailed type of feedback than those who passively watch from their seats or do not pay attention to the checking process.

Moreover, many elementary and some middle school teachers in all focus group discussions mentioned asking questions, or assigning exercises similar to those of previous homework assignments (see Figure 1 , Testing of related content). Data from the classroom observations confirmed this practice. Participants stressed that this practice provides students with a new feedback event centered on the level of accuracy of their responses and on their ability to transfer the knowledge learned to new tasks. However, despite the general agreement regarding this homework feedback practice, some middle school teachers admitted that they only check students’ ability to transfer knowledge in assessment tests and claimed that this practice should not be considered homework feedback – “This is assessment, not feedback! [Emphatic tone]” (F1P5).

Most participants at both school levels reported following their school’s assessment criteria regarding homework. Generally, homework completion counts for 2–5% of the overall grade in mathematics. When asked to be more specific, several teachers explained that they use information on homework completion recorded in class logs, while others declared using information on students’ performance when checking homework on the board. Teachers admitted that they do not examine the quality of all homework assignments given in class because of the heavy workload they faced on a daily basis. During classroom observations, teachers registered who did not complete homework and sometimes they referred that this behavior would decrease their overall grade. Most teachers in all focus groups reported including information on homework completion in the overall mathematics grade; still, less than half identified this practice as a type of homework feedback.

Furthermore, some of the elementary and middle school teachers in all focus groups mentioned sending parents a message when their children miss homework three times as a type of homework feedback. This practice was confirmed by data from classroom observations. Interestingly, participants did not mention reporting children’s progress on homework to parents during the focus group discussions, and accordingly this practice was not observed in class.

Finally, a few elementary school teachers in two focus group discussions and a few middle school teachers in one focus group reported commenting on students’ homework regularly. Comments address the strengths and weaknesses of homework, pointing out the topics that need to be improved, as the following quotation exemplifies:

F4P1: I comment on what is done well, but I also point out mistakes and suggest ways to improve what is wrong or not so well done. For example, I’d write: “Great line of reasoning but try to do x so you’ll only have to do two calculations and you’ll finish the exercises faster.” Unfortunately, sometimes I have to write other kinds of comments such as “What a coincidence, your answer is exactly the same as Joana’s or Catarina’s … and the three of you have made exactly the same mistakes…”

These few participants were asked by their focus group peers how they managed to comment on students’ homework regularly. A teacher answered that she could do it because she had been assigned only one class; still, “I spent my lunch hour and some of my free time at school working on this” (F3P3). Another teacher explained that she provides this type of homework feedback weekly, except for those weeks when students have assessment tests. According to this last participant, the negative side of this practice, “frustrating I should say” (F4P1), is when students copy homework answers from another student. Commenting on students’ homework is a very time-consuming practice, and this participant expressed feeling discouraged when such maladaptive behaviors occur in class. To overcome the “very time-consuming obstacle,” another teacher who also claimed to use this practice explained that he usually asks the whole class to complete homework on a separate sheet – “I choose only one good exercise which reflects the material covered in class” (F5P7). In the next lesson, and without prior notice, he collects four or five homework assignments, which are returned with feedback comments in the following class. Participants in the three focus group discussions agreed that this type of homework feedback is very useful, but also stressed the unlikelihood of giving it in class because of the heavy workload they as teachers have to bear (e.g., teachers have to teach five or six classes at different grade levels, each of them with over 25 students, heavy curriculums). In this context, one participant complained: “I’m not a rubber band that may be stretched [endlessly]” (F5P5).

Perceived Impact of Homework Feedback

As Figure 1 shows, elementary and middle school teachers highlighted the positive impact of homework feedback on content learning, self-esteem, and homework completion (some categories are typical and others are variant).

The following dialog among elementary school teachers illustrates their conceptions on the impact of homework feedback:

F4P9: Students who complete homework regularly are more willing to understand the contents covered.

F4P2: …and they complete homework more often… At least I notice more effort.

Moreover, both elementary and middle school teachers related homework feedback to class participation (variant category in both school levels), as the following participant argued:

F4P5: Yes…they [students who complete homework regularly] follow classroom instructions and participate in class more actively, for example, by asking me questions and answering mine more frequently…

Only elementary school teachers in two focus group discussions related homework feedback to students’ achievement, while none of the middle school teachers did so (see Figure 1 ). In fact, some of the middle school teachers in all the focus group discussions defended the need for students to play an active role in their learning, arguing that homework feedback is not worthwhile for those who are not interested in learning.

Relationships between Teachers’ Conceptions of Homework Feedback

The second research question aimed to examine how the four key aspects of the homework feedback are related. Figure 3 provides a graphical model of teachers’ most salient conceptions of homework feedback and the relationships among them. The bold solid lines represent typical cases (more than 50%), the thinner solid lines represent variant categories (between 25 and 50% cases), and the dotted lines represent variant categories (between 17 and 24% cases). All lines represent the conceptions of both elementary and middle school teachers except for the lines with an asterisk, which refer to a specific school level (see legend of Figure 3 ).

As Figure 3 shows, the definitions of homework feedback provided by elementary and middle school teachers differ regarding the purposes for giving homework feedback. The middle school teachers perceived homework feedback as the feedback provided by the teacher to their students about their homework. The purpose for this homework feedback was described by teachers as twofold: help teachers monitor students’ learning and help students monitor their own learning. The latter was mentioned less often by middle school teachers. Besides, the middle school teachers conceptualized homework feedback provided to teachers by their students with the purpose of helping teachers monitor students’ learning.

In turn, elementary school teachers perceived homework feedback mainly as self-feedback and, accordingly, conceptualized students’ monitoring of their learning as the main purpose for giving homework feedback. While discussing, these teachers highlighted students’ active role in self-regulate their learning during and after homework completion (e.g., students checking their answers when solutions are written on the board). Still, the elementary school teachers did not explain how they promote these self-regulation skills in class. Moreover, the second set of relationships (i.e., purposes and homework feedback types) reveals a different pattern of results as described below.

Interestingly, the participating teachers operationalized both homework purposes (i.e., teachers monitoring students learning and students’ monitoring their own learning, the latter less often; see Figure 3 ) through the “checking homework on the board” homework feedback type. Teachers’ arguments were twofold: this practice allows checking students’ level of understanding of content (e.g., students solving exercises autonomously on the board), and students can learn about their skills while checking their answers with those written on the board.

The homework feedback practice testing related contents was also linked to both purposes but only by elementary school teachers. These teachers argued that providing students with similar exercises to those previously set as homework helps teachers monitor their students’ learning and students to monitor their own learning.

The purpose “teachers’ monitoring their students’ learning” was linked to the practice “checking homework completion” by elementary and middle school teachers. This homework feedback practice helps teachers learn who completed homework and collect information on the content with which students are struggling the most. This information is expected to help teachers meet their students’ needs.

Finally, the various types of homework feedback were associated with various perceived impacts. Teachers at both school levels converged in the fact that checking homework completion impacts students’ homework completion positively. In general, teachers mentioned that some students are “immature” and their lack of active involvement and strong volition prevent them from completing homework. Thus, most of the teachers at both school levels anticipated that external control is needed to help students complete homework. Checking homework completion was referred to as an important tool for encouraging students to do homework.

As Figure 3 depicts, teachers described checking homework on the board as the homework feedback practice that most benefits students. According to participants, this practice fosters self-esteem (only reported by elementary school teachers), homework completion (reported by some teachers at both school levels), class participation (reported by some teachers at both levels), and learning of the content taught in class (reported by most of the teachers from both levels). Teachers explained that praising students on their good performance while doing exercises on the board is likely to increase their self-esteem. Furthermore, teachers said that this practice encourages homework completion and increases class participation because it provides students with specific information on how to solve exercises.

Some elementary school teachers reported that testing related content helps students participate more in class (e.g., answering teacher’s questions, asking more questions) and be more engaged in their learning.

A few teachers at both school levels (see Figure 3 ) mentioned counting homework in the overall grade, and communicating with parents when their children miss homework three times as two types of homework feedback with impact on students’ homework completion. Counting homework completion in the overall grade was referred to as a direct incentive for students to complete homework. However, some teachers alerted that this practice may not always be effective because of the time gap between students’ homework behaviors and the end of term when they get their final grade report. Thus, all agreed that teachers should respond to students’ homework behaviors (e.g., missing homework or doing assignments correctly) as soon as possible. Participants highlighted the importance of communicating with families about children’s homework behaviors. However, teachers alerted that this type of homework feedback may not be effective without the implication of the family in the learning process; “if the family is aware of the importance of this type of practice, then it will be effective, otherwise it will have no effect” (F3P4).

As reported previously, 17% of the elementary and middle school teachers claimed to make written comments on students’ homework assignments (see Figure 1 ). However, when discussing the possible impact of the various types of homework feedback, more teachers (than that 17%) agreed that written comments on students’ assignments would improve students’ learning of content (see Figure 3 ). These teachers mentioned that personalized homework feedback would help students correct their mistakes and might provide guidance on the topics that need to be further studied. As a result, students were likely to improve their grades.

The discussion of the current study is organized according to each key aspect of teachers’ conceptions of homework feedback. Regarding the first key aspect of homework feedback, teachers proposed a multifaceted definition of homework feedback: (i) homework feedback provided by the teacher, (ii) homework feedback provided by the student, and (iii) homework self-feedback. The latter extends the definition of Cooper (2001) , who defined homework feedback as the teachers’ responses to students’ homework completion as a follow-up (e.g., comments, incentives, grades). The definition of homework self-feedback is linked to the internal feedback or self-feedback proposed by Butler and Winne (1995) and Hattie and Timperley (2007) , respectively. According to these authors, students are expected to display self-regulatory skills to self-evaluate their performance in homework assignments (see Hattie and Timperley, 2007 ). Interestingly, this category is typical in elementary school, but variant in middle school. This is an important finding because the generation of internal feedback requires knowledge on strategies and standards, as well as the capacity to judge the quality of a task in relation to standards, which not all students are capable of, especially those at lower grade levels ( Zimmerman and Martinez-Pons, 1990 ; Butler and Winne, 1995 ; Rosário et al., 2016 ). Moreover, low achievers struggling to learn often fail to activate and control the SRL process ( Núñez et al., 2015a ). In fact, these students are likely to fail to monitor their homework behaviors because they do not know “whether they are on the right track” (F4P8).

Consistently with literature, teachers’ major conceptions of homework feedback purposes addressed monitoring students’ learning, either focusing on teachers’ or on students’ role ( Corno, 2000 ; An and Wu, 2012 ; Bang, 2012 ). This may be particularly important in mathematics where contents are organized so as to follow a continuous progression and lower levels prepare the foundations of subsequent levels ( Pijls and Dekker, 2011 ). Teachers’ monitoring provides the opportunity for teachers to change their teaching practices in response to students’ needs ( Walberg and Paik, 2000 ; Kralovec and Buell, 2001 ), which may be understood as a “student-centered” approach (see Sheridan, 2013 ). The conception of homework feedback purposes focused on students’ monitoring their work emphasizes students’ active role during the homework process and the use of SRL competencies such as self-monitoring and self-reflection (e.g., Ramdass and Zimmerman, 2011 ; Zhu and Leung, 2012 ). The last purpose of homework feedback proposed by participants is to “promote self-esteem.” This purpose is not mentioned in homework literature; however, in the study by Irving et al. (2011) , teachers mentioned the need to inform students about the positive aspects of their performance, thus incentivizing their progress, especially among low achievers showing low self-esteem.

Regarding the third topic of homework feedback (homework feedback types), findings in the current study are consistent with literature ( Cooper, 2001 ; Mullis et al., 2004 ; Kaur, 2011 ; Zhu and Leung, 2012 ; Rosário et al., 2015b ; Kukliansky et al., 2016 ). However, despite the similarity of the homework feedback practices reported by elementary and middle school teachers, the percentages of each reported category vary. For example, checking homework completion and checking homework on the board are general categories in elementary school and typical categories in middle school; while testing of related content is a typical category in elementary school and a variant category in middle school. These findings are consistent with students’ reports on their teachers’ support in homework ( Katz et al., 2010 ; Núñez et al., 2015b ). A decrease in teachers’ support in homework at middle school level is expected because older students are likely to be more autonomous. However, Katz et al. (2010) found that the middle school students who perceived low teachers’ homework support reported high psychological needs and low intrinsic motivation. Other important finding to note is the use of criticism observed in elementary and middle school classrooms which may have the opposite effect of teachers’ intentions (e.g., reduce homework non-compliance). In fact, being criticized in class is likely to be non-constructive because it may reduce students’ willingness to accept criticism and result in low favorable responses toward homework. On the contrary, criticism delivered in private is likely to lead to better responses (see Leung et al., 2001 ).

According to participants, homework feedback impacts in the following aspects: content learning, self-esteem, homework completion, class participation, and achievement. Globally, this finding is consistent with previous research (e.g., Trautwein and Lüdtke, 2009 ; Xu, 2011 ; Núñez et al., 2015b ), except for class participation and self-esteem which have not yet been studied. It is interesting to note, however, that despite most teachers reported spending 30 min or more providing homework feedback in each class (see Initial Data Screening subsection); about one third of elementary school teachers related homework feedback to students’ achievement, while none of the middle school teachers did so. However, prior research has evidenced the positive impact of homework feedback on students’ academic achievement ( Núñez et al., 2015b ), especially when teachers provide suggestions on how to improve learning (see Elawar and Corno, 1985 ; Walberg et al., 1985 ; Rosário et al., 2015b ).

Moreover, middle school teachers added that when students do not play an active role in their learning, feedback is not likely to have any impact. This conception is consistent with the SRL approach to the homework process (e.g., Xu and Wu, 2013 ; Xu, 2014 ; Núñez et al., 2015b ) which stresses, for example, the role that teachers may play in helping students define their own homework goals and reflect on the relationship between homework completion and achieving self-set learning goals (e.g., Núñez et al., 2015b ; Rosário et al., 2015b ). As Labuhn et al. (2010) observed, the feedback provided by teachers may not impact students’ learning and behaviors if students do not understand what is intended with homework feedback.

Findings gathered from relationships between teachers’ conceptions of homework feedback provide additional useful insights. Interestingly, the two most frequently reported types of homework feedback (i.e., checking homework completion and checking homework on the board) in both school levels are more linked to the purpose “teachers monitoring students’ learning” than to the purpose “students monitoring their own learning.” This data may suggest that teachers may not be fully aware of the importance of promoting students’ SRL competencies to increase the benefits of homework feedback or they may lack the knowledge to promote these skills in class (see Spruce and Bol, 2015 ).

Practical Implications

The current study provides four major findings of relevance for educational practice: (i) decrease of teachers’ reported homework feedback practices from elementary to middle school level; (ii) a few teachers from elementary school and none from middle school level perceive homework feedback impacting on students’ academic achievement; (iii) usage of public criticism in class, especially in middle school; and (iv) teachers’ lack of awareness on SRL strategies.

First, teachers and school administrators with the help of school psychologists could examine homework practices delivered in class, namely homework feedback, to analyze whether they are set to be responsive to students’ educational needs. As found in the current study, there is a decrease of the homework feedback from elementary to middle school; however, this finding should be considered by teachers because, according to literature, many middle school students still report the need of teachers’ homework support (e.g., Katz et al., 2010 ).

Data also showed that both elementary and middle school teachers spend around 30 min providing homework feedback in class, but the perceived impact of this school practice on students’ achievement was barely mentioned in the focus groups. This data merit reflection within the school context to understand whether homework feedback is being used with efficacy. For example, school-based training for teachers’ on homework models (e.g., Cooper, 2001 ; Trautwein et al., 2006 ) could theoretically ground their homework practices in schools. This training would also help teachers understand that criticism and irony in class may discourage homework compliance, but it also may lead to undesirable outcomes such as children homework disengagement.

Finally, data (e.g., elementary school teachers believe that students generate homework self-feedback; the homework feedback practices most used in class are more closely related to the purpose “teachers monitoring students’ learning” than to the purpose “students monitoring their own learning”) suggest the need to set school-based training for teachers on SRL strategies. This training could consider addressing the homework process in relation with SRL to promote students’ agency and sense of responsibility over homework and homework feedback in particular. For example, teachers are expected to learn and practice how to model the use of SRL strategies in class ( Rosário et al., 2013 ; Spruce and Bol, 2015 ). In fact, students lacking SRL skills may fail to use the homework feedback delivered in class, which may compromise the impact of this instructional tool on students learning and achievement (see Corno, 2000 ; Peterson and Irving, 2008 ; Zhu and Leung, 2012 ). To promote the development of students SRL competencies and increase the benefits of homework feedback, teachers may also consider using “diary tasks” to promote students’ homework self-reflection in class (see Ferreira et al., 2015 ).

Strengths, Limitations, and Future Research

To authors’ knowledge, this study was the first to map mathematics teachers’ conceptions of homework feedback and examine the relationships between teachers’ definitions, purposes, types, and perceived impact of homework feedback. The analysis of these relationships focusing on a specific content domain at two school levels showed which categories were linked, and how, by the participating teachers. This study extended previous research conducted with mathematics teachers from a single grade level (see Kaur, 2011 ; Zhu and Leung, 2012 ).

According to the current findings, elementary and middle school teachers’ conceptions of homework feedback vary, as well as the time spent in class providing feedback. Moreover, in spite of the fact that the types of homework feedback practices are the same, the type of categories (i.e., general vs. typical and typical vs. variant) varies in the two school levels, and the dynamic of providing homework feedback at those school levels is diverse and complex (e.g., usage of various strategies to provide some types of homework feedback, even by the same teacher). These findings may help understand why the relationship between homework and academic achievement reported in the literature varies from elementary to middle school (see Cooper et al., 2006 ; Fan et al., 2017 ).

Furthermore, the complexity of the homework feedback process reflected by the collected data may not be captured by extant instruments that examine teachers’ homework feedback practices. To some extent, this may contribute to understand the low effect sizes and explained variances found in the homework feedback research (e.g., Xu, 2014 ; Rosário et al., 2015b ). This finding reinforces the need for future studies collect data using more than one method to capture and better understand the phenomenon of the homework process and its influence on students’ academic outcomes (e.g., Cooper et al., 2006 ). Furthermore, findings showed positive relationships between some types of homework feedback practices and perceived impact on students’ variables that have not yet been examined in homework research (e.g., checking homework on the blackboard and class participation). Future studies may consider further examining these relationships.

The present study followed methodological procedures to enhance trustworthiness of findings such as random sampling, investigator and methodological triangulation, provision of direct quotations, and member checking ( Lincoln and Guba, 1985 ; Shenton, 2004 ; Elo et al., 2014 ). Results from member checking were very positive. The majority of the participants agreed that the description of the findings was a genuine reflection of the topics covered in the focus group discussions, and of the homework routines in the classroom. No suggestions were made to change the description of data. Such data have strengthened present findings. In addition, teachers highlighted that they usually choose types of homework feedback that reach all students because of the professional constraints they experience daily (i.e., heavy workload). This topic was mentioned during the discussions and may merit further investigation because it may be an important factor compromising the homework feedback process.

Notwithstanding the strengths of the current study, there are also some limitations that need to be addressed. Classroom observations helped strengthen findings, nevertheless only 25% of the participating teachers were observed in a limited period of time. Moreover, most of the participants have extensive experience in teaching, which may have contributed to the results. As Hattie (2003) reported, expert teachers are more capable of seeking and giving feedback, and also monitoring their students’ learning than novice teachers. Conducting studies on novice teachers would help identify their specific needs for training on instructional variables, and design school-based interventions to meet these professionals’ needs.

Elementary and middle school teachers’ conceptions of homework feedback were mapped, but the role of students in the homework feedback process should be further researched. Further investigation may want to explore elementary and middle school students’ conceptions of homework feedback and compare their responses with current findings. The information provided would be useful to learn how students understand (e.g., in what ways students perceive teachers’ homework feedback practices as helpful, see Xu, 2016 ) and cope with the homework feedback given in class. Examining the (mis)alignment of both conceptions of homework feedback (elementary and middle school teachers and students) may help deepen the understanding of the impact of homework feedback and further examine the differential relationship between doing homework and academic achievement at these two school levels (see Cooper et al., 2006 ). The results, although promising, should be further investigated in different school grades and subjects. At this level, however, they may be useful to researchers looking for an in-depth understanding of homework feedback and willing to explore new research topics on the “last but not least” aspect of the homework process.

Ethics Statement

This study was reviewed and approved by the ethics committee of the University of Minho. All research participants provided written informed consent in accordance with the Declaration of Helsinki.

Author Contributions

JC and PR substantially contributed to the conception and the design of the work. JC was responsible for the literature search. JC, AN, TM, JN, and TN were responsible for the acquisition, analysis, and interpretation of data for the work. PR was also in charge of technical guidance. JN made important intellectual contribution in manuscript revision. JC wrote the manuscript with valuable inputs from the remaining authors. All authors agreed for all aspects of the work and approved the version to be published.

This study was conducted at Psychology Research Centre (UID/PSI/01662/2013), University of Minho, and supported by the Portuguese Foundation for Science and Technology and the Portuguese Ministry of Science, Technology and Higher Education through national funds and co-financed by FEDER through COMPETE2020 under the PT2020 Partnership Agreement (POCI-01-0145-FEDER-007653). JC was supported by a Ph.D. fellowship from the Portuguese Foundation for Science and Technology (FCT – SFRH/BD/95341/2013).

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

Authors would like to thank Sofia Kirkman and Fuensanta Monroy for the English editing of the manuscript.

An, S., and Wu, Z. (2012). Enhancing mathematics teachers’ knowledge of students’ thinking from assessing and analyzing misconceptions in homework. Int. J. Sci. Math. Educ. 10, 717–753. doi: 10.1007/s10763-011-9324-x

CrossRef Full Text | Google Scholar

Bang, H. (2012). Promising homework practices: teachers’ perspectives on making homework work for newcomer immigrant students. High Sch. J. 95, 3–31. doi: 10.1353/hsj.2012.0001

Bang, H., Suárez-Orozco, C., Pakes, J., and O’Conner, E. (2009). The importance of homework in determining immigrant students’ grades in schools in the USA context. Educ. Res. 51, 1–25. doi: 10.1080/00131880802704624

Bang, H. J. (2011). What makes it easy or hard for you to do your homework? An account of newcomer immigrant youths’ afterschool academic lives. Curr. Issues Educ. 14, 1–26.

Google Scholar

Bardin, L. (1996). The Content Analysis. Paris: PUF.

Bazeley, P., and Jackson, K. (2013). Qualitative Data Analysis with NVivo. London: Sage.

Bedford, P. D. (2014). Teachers’ Beliefs and Practices Regarding Homework: An Examination of the Cognitive Domain Embedded in Third Grade Mathematics Homework. Doctoral dissertation, University of Wisconsin-Milwaukee, Milwaukee, WI.

Brookhart, S. M. (2008). How to Give Effective Feedback to Your Students. Alexandria, VA: ASCD.

Butler, D. L., and Winne, P. H. (1995). Feedback and self-regulated learning: a theoretical synthesis. Rev. Educ. Res. 65, 245–281. doi: 10.3102/00346543065003245

Cardelle, M., and Corno, L. (1981). Effects on second language learning of variations in written feedback on homework assignments. TESOL Q. 15, k251–261. doi: 10.2307/3586751

Choo, E. K., Garro, A. C., Ranney, M. L., Meisel, Z. F., and Morrow Guthrie, K. (2015). Qualitative research in emergency care part I: research principles and common applications. Acad. Emerg. Med. 22, 1096–1102. doi: 10.1111/acem.12736

PubMed Abstract | CrossRef Full Text | Google Scholar

Cooper, H. (2001). The Battle Over Homework: Common Ground for Administrators, Teachers, and Parents , 2nd Edn. Thousand Oaks, CA: Sage Publications.

Cooper, H., Robinson, J., and Patall, E. (2006). Does homework improve academic achievement? A synthesis of research. Rev. Educ. Res. 76, 1–62. doi: 10.3102/00346543076001001

Cooper, H., and Valentine, J. C. (2001). Using research to answer practical questions about homework. Educ. Psychol. 36, 143–153. doi: 10.1207/S15326985EP3603_1

Corno, L. (2000). Looking at homework differently. Elem. Sch. J. 100, 529–548. doi: 10.1086/499654

Creswell, J. W. (2007). Qualitative Inquiry and Research Method: Choosing Among Five Approaches , 2nd Edn. Thousand Oaks, CA: Sage.

Deslandes, R. (2009). “Elementary school teachers’ views of homework and parents - school relations,” in Perspectives on Student Outcomes and Homework: Family-School-Community Partnerships , ed. R. Deslandes (New York, NY: Routledge), 128–140.

Dettmers, S., Trautwein, U., Lüdtke, O., Kunter, M., and Baumert, J. (2010). Homework works if homework quality is high: using multilevel modeling to predict the development of achievement in mathematics. J. Educ. Psychol. 102, 467–482. doi: 10.1037/a0018453

Elawar, M. C., and Corno, L. (1985). A factorial experiment in teachers’ written feedback on student homework: changing teacher behavior a little rather than a lot. J. Educ. Psychol. 77, 162–173. doi: 10.1037/0022-0663.77.2.162

Elo, S., Kääriäinen, M., Kanste, O., Pölkki, T., Utriainen, K., and Kyngäs, H. (2014). Qualitative content analysis: a focus on trustworthiness. SAGE Open 4, 1–10. doi: 10.1177/2158244014522633

Epstein, J. L., and Van Voorhis, F. L. (2001). More than ten minutes: teachers’ roles in designing homework. Educ. Psychol. 36, 181–193. doi: 10.1207/S15326985EP3603_4

Fan, H., Xu, J., Cai, Z., He, J., and Fan, X. (2017). Homework and students’ achievement in math and science: a 30-year meta-analysis, 1986-2015. Educ. Res. Rev. 20, 35–54. doi: 10.1016/j.edurev.2016.11.003

Fernández-Alonso, R., Suárez-Álvarez, J., and Muñiz, J. (2015). Adolescents’ homework performance in mathematics and science: personal factors and teaching practices. J. Educ. Psychol. 107, 1075–1085. doi: 10.1037/edu0000032

Ferreira, P. C., Simão, A. V., and Silva, A. L. (2015). Does training in how to regulate one’s learning affect how students report self-regulated learning in diary tasks? Metacogn. Learn. 10, 199–230. doi: 10.1007/s11409-014-9121-3

Hagger, M., Sultan, S., Hardcastle, S., and Chatzisarantis, N. (2015). Perceived autonomy support and autonomous motivation toward mathematics activities in educational and out-of-school contexts is related to mathematics homework behavior and attainment. Contemp. Educ. Psychol. 41, 111–123. doi: 10.1016/j.cedpsych.2014.12.002

Hattie, J., and Timperley, H. (2007). The power of feedback. Rev. Educ. Res. 77, 81–112. doi: 10.3102/003465430298487

Hattie, J. A. (2003). Teachers make a difference: what is the research evidence?. Paper Presented at the Building Teacher Quality Research Conference , Melbourne, VIC. Available at: https://research.acer.edu.au/cgi/viewcontent.cgi?article=1003&context=research_conference_2003

Hill, C. E., Knox, S., Thompson, B. J., Williams, E. N., Hess, S. A., and Ladany, N. (2005). Consensual qualitative research: an update. J. Couns. Psychol. 52, k196–205. doi: 10.1037/a0033361

Irving, S., Harris, L., and Peterson, E. (2011). ‘One assessment doesn’t serve all the purposes’ or does it? New Zealand teachers describe assessment and feedback. Asia Pac. Educ. Rev. 12, 413–426. doi: 10.1007/s12564-011-9145-1

Katz, I., Kaplan, A., and Gueta, G. (2010). Students’ needs, teachers’ support, and motivation for doing homework: a cross-sectional study. J. Exp. Educ. 78, 246–267. doi: 10.1080/00220970903292868

Kaur, B. (2011). Mathematics homework: a study of three grade eight classrooms in Singapore. Int. J. Sci. Math. Educ. 9, 187–206. doi: 10.1007/s10763-010-9237-0

Kim, S. Y., Graham, S. S., Ahn, S., Olson, M. K., Card, D. J., Kessler, M. M., et al. (2016). Correcting biased Cohen’s Kappa in NVivo. Commun. Methods Meas. 10, 217–232. doi: 10.1080/19312458.2016.1227772

Kitzinger, J. (1995). Qualitative research: introducing focus groups. BMJ 311, 299–302. doi: 10.1136/bmj.311.7000.299

Kralovec, E., and Buell, J. (2001). End homework now. Educ. Leadersh. 58, 39–42.

PubMed Abstract | Google Scholar

Krueger, R. A., and Casey, M. A. (2000). Focus Groups: A Practical Guide for Applied Research , 3rd Edn. Thousand Oaks, CA: Sage. doi: 10.1037/10518-189

Krueger, R. A., and Casey, M. A. (2010). “Focus group interviewing,” in Handbook of Practical Program Evaluation , 3rd Edn, eds K. E. Newcomer, H. P. Hatry, and J. S. Wholey (San Francisco, CA: Jossey-Bass), 506–534.

Kukliansky, I., Shosberger, I., and Eshach, H. (2016). Science teachers’ voice on homework: beliefs, attitudes, and behaviors. Int. J. Sci. Math. Educ. 14, 229–250. doi: 10.1007/s10763-014-9555-8

Labuhn, A. S., Zimmerman, B. J., and Hasselhorn, M. (2010). Enhancing students’ self-regulation and mathematics performance: the influence of feedback and self-evaluative standards. Metacogn. Learn. 5, 173–194. doi: 10.1007/s11409-010-9056-2

Landis, J. R., and Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics 33, 159–174. doi: 10.2307/2529310

Leung, K., Su, S., and Morris, M. W. (2001). When is criticism not constructive? The roles of fairness perceptions and dispositional attributions in employee acceptance of critical supervisory feedback. Hum. Relat. 54, 1155–1187. doi: 10.1177/0018726701549002

Lincoln, Y. S., and Guba, E. G. (1985). Naturalistic Inquiry. Beverly Hills, CA: Sage.

Morgan, D. L. (1997). Focus Group as Qualitative Research , 2nd Edn. Thousand Oaks, CA: Sage. doi: 10.4135/9781412984287

Muhlenbruck, L., Cooper, H., Nye, B., and Lindsay, J. J. (1999). Homework and achievement: explaining the different strengths of relation at the elementary and secondary school levels. Soc. Psych. Educ. 3, 295–317. doi: 10.1023/A:1009680513901

Mullis, I. V., Martin, M. O., Gonzalez, E. J., and Chrostowski, S. J. (2004). TIMSS 2003 International Mathematics Report: Findings from IEA’s Trends in International Mathematics and Science Study at the Fourth and Eighth Grades. Chestnut Hill, MA: Boston College.

Murphy, J., Decker, K., Chaplin, C., Dagenais, R., Heller, J., Jones, R., et al. (1987). An exploratory analysis of the structure of homework assignments in high schools. Res. Rural Educ. 4, 61–71.

Núñez, J. C., Rosário, P., Vallejo, G., and González-Pienda, J. (2013). A longitudinal assessment of the effectiveness of a school-based mentoring program in middle school. Contemp. Educ. Psychol. 38, 11–21. doi: 10.1016/j.cedpsych.2012.10.002

Núñez, J. C., Suárez, N., Cerezo, R., González-Pienda, J., Rosário, P., Mourão, R., et al. (2015a). Homework and academic achievement across Spanish compulsory education. Educ. Psychol. 35, 726–746. doi: 10.1080/01443410.2013.817537

Núñez, J. C., Suárez, N., Rosário, P., Vallejo, G., Cerezo, R., and Valle, A. (2015b). Teachers’ feedback on homework, homework-related behaviors, and academic achievement. J. Educ. Res. 108, 204–216. doi: 10.1080/00220671.2013.878298

Núñez, J. C., Suárez, N., Rosário, P., Vallejo, G., Valle, A., and Epstein, J. L. (2015c). Relationships between parental involvement in homework, student homework behaviors, and academic achievement: differences among elementary, junior high, and high school students. Metacogn. Learn. 10, 375–406. doi: 10.1007/s11409-015-9135-5

OECD (2014). PISA 2012 Results in Focus: What 15-Year-Olds Know and what they can do with what they know, PISA. Paris: OECD Publishing.

Peterson, E., and Irving, S. (2008). Secondary school students’ conceptions of assessment and feedback. Learn. Instr. 18, 238–250. doi: 10.1016/j.learninstruc.2007.05.001

Pijls, M., and Dekker, R. (2011). Students discussing their mathematical ideas: the role of the teacher. Math. Educ. Res. J. 23, 379–396. doi: 10.1007/s13394-011-0022-3

Ramdass, D., and Zimmerman, B. J. (2011). Developing self-regulation skills: the important role of homework. J. Adv. Acad. 22, 194–218. doi: 10.1177/1932202X1102200202

Reyna, V. F., and Brainerd, C. J. (2007). The importance of mathematics in health and human judgment: numeracy, risk communication, and medical decision making. Learn. Individ. Diff. 17, 147–159. doi: 10.1016/j.lindif.2007.03.010

Richards, L. (2005). Handling Qualitative Data: A Practical Guide. London: Sage Publications.

Rosário, P., Núñez, J. C., Valle, A., González-Pienda, J., and Lourenço, A. (2013). Grade level, study time, and grade retention and their effects on motivation, self-regulated learning strategies, and mathematics achievement: a structural equation model. Eur. J. Psychol. Educ. 28, 1311–1331. doi: 10.1007/s10212-012-0167-9

Rosário, P., Núñez, J. C., Vallejo, G., Cunha, J., Azevedo, R., Pereira, R., et al. (2016). Promoting Gypsy children school engagement: a story-tool project to enhance self-regulated learning. Contemp. Educ. Psychol. 47, 84–94. doi: 10.1016/j.cedpsych.2015.11.005

Rosário, P., Núñez, J. C., Vallejo, G., Cunha, J., Nunes, T., Mourão, R., et al. (2015a). Does homework design matter? The role of homework’s purpose in student mathematics achievement. Contemp. Educ. Psychol. 43, 10–24. doi: 10.1016/j.cedpsych.2015.08.001

Rosário, P., Núñez, J. C., Vallejo, G., Cunha, J., Nunes, T., Suárez, N., et al. (2015b). The effects of teachers’ homework follow-up practices on students’ EFL performance: a randomized-group design. Front. Psychol. 6:1528. doi: 10.3389/fpsyg.2015.01528

Ryan, K. E., Gandha, T., Culbertson, M. J., and Carlson, C. (2014). Focus group evidence: implications for design and analysis. Am. J. Eval. 35, 328–345. doi: 10.1177/1098214013508300

Schmitz, B., and Perels, F. (2011). Self-monitoring of self-regulation during math homework behaviour using standardized diaries. Metacogn. Learn. 6, 255–273. doi: 10.1007/s11409-011-9076-6

Shenton, A. (2004). Strategies for ensuring trustworthiness in qualitative research projects. Educ. Inf. 22, 63–75. doi: 10.3233/EFI-2004-22201

Sheridan, L. D. (2013). Changes in pre-service teachers perceptions’ of teacher qualities: development from egocentric to student centric. Aust. J. Teach. Educ. 38, 55–75. doi: 10.14221/ajte.2013v38n9.2

Spruce, R., and Bol, L. (2015). Teacher beliefs, knowledge, and practice of self-regulated learning. Metacogn. Learn. 10, 245–277. doi: 10.1007/s11409-014-9124-0

Tas, Y., Sungur, S., and Oztekin, C. (2016). Development and validation of science homework scale for middle-school students. Int. J. Sci. Math. Educ. 14, 417–444. doi: 10.1007/s10763-014-9582-5

Trautwein, U., and Lüdtke, O. (2009). Predicting homework motivation and homework effort in six school subjects: the role of person and family characteristics, classroom factors, and school track. Learn. Instr. 19, 243–258. doi: 10.1016/j.learninstruc.2008.05.001

Trautwein, U., Lüdtke, O., Schnyder, I., and Niggli, A. (2006). Predicting homework effort: support for a domain-specific, multilevel homework model. J. Educ. Psychol. 98, 438–456. doi: 10.1037/0022-0663.98.2.438

Walberg, H. J., and Paik, S. J. (2000). Effective Educational Practices. Brussels: International Academy of Education & International Bureau of Education.

Walberg, H. J., Paschal, R. A., and Weinstein, T. (1985). Homework’s powerful effects on learning. Educ. Leadersh. 42, 76–79.

Xu, J. (2008). Models of secondary school students’ interest in homework: a multilevel analysis. Am. Educ. Res. J. 45, 1180–1205. doi: 10.3102/0002831208323276

Xu, J. (2010). Homework purposes reported by secondary school students: a multilevel analysis. J. Educ. Res. 103, 171–182. doi: 10.1080/00220670903382939

Xu, J. (2011). Homework completion at the secondary school level: a multilevel analysis. J. Educ. Res. 104, 171–182. doi: 10.1080/00220671003636752

Xu, J. (2012). Predicting students’ homework environment management at the secondary school level. Educ. Psychol. 32, 183–200. doi: 10.1080/01443410.2011.635639

Xu, J. (2014). Regulation of motivation: predicting students’ homework motivation management at the secondary school level. Res. Pap. Educ. 29, 457–478. doi: 10.1080/02671522.2013.775324

Xu, J. (2015). Investigating factors that influence conventional distraction and tech-related distraction in math homework. Comput. Educ. 81, 304–314. doi: 10.1016/j.compedu.2014.10.024

Xu, J. (2016). A study of the validity and reliability of the teacher homework involvement scale: a psychometric evaluation. Measurement 93, 102–107. doi: 10.1016/j.measurement.2016.07.012

Xu, J., Du, J., and Fan, X. (2017). Self-regulation of mathematics homework behavior: an empirical investigation. J. Educ. Res. 110, 467–477. doi: 10.1080/00220671.2015.1125837

Xu, J., and Wu, H. (2013). Self-regulation of homework behavior: homework management at the secondary school level. J. Educ. Res. 106, 1–13. doi: 10.1080/00220671.2012.658457

Zhu, Y., and Leung, F. (2012). Homework and mathematics achievement in Hong Kong: evidence from the TIMSS 2003. Int. J. Sci. Math. Educ. 10, 907–925. doi: 10.1007/s10763-011-9302-3

Zimmerman, B. J. (1989). A social cognitive view of self-regulated academic learning. J. Educ. Psychol. 81, 329–339. doi: 10.1037/0022-0663.81.3.329

Zimmerman, B. J., and Kitsantas, A. (2005). Homework practices and academic achievement: the mediating role of self-efficacy and perceived responsibility beliefs. Contemp. Educ. Psychol. 30, 397–417. doi: 10.1016/j.cedpsych.2005.05.003

Zimmerman, B. J., and Martinez-Pons, M. (1990). Student differences in self-regulated learning: relating grade, sex, and giftedness to self-efficacy and strategy use. J. Educ. Psychol. 82, 51–59. doi: 10.1037/0022-0663.82.1.51

Keywords : homework feedback, teachers’ conceptions, homework feedback purposes, perceived impact, focus group, classroom observations

Citation: Cunha J, Rosário P, Núñez JC, Nunes AR, Moreira T and Nunes T (2018) “Homework Feedback Is…”: Elementary and Middle School Teachers’ Conceptions of Homework Feedback. Front. Psychol. 9:32. doi: 10.3389/fpsyg.2018.00032

Received: 13 September 2017; Accepted: 10 January 2018; Published: 06 February 2018.

Reviewed by:

Copyright © 2018 Cunha, Rosário, Núñez, Nunes, Moreira and Nunes. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pedro Rosário, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

More from M-W
To save this word, you'll need to log in. Log In

Definition of homework

Examples of homework in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'homework.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

1662, in the meaning defined at sense 1

Dictionary Entries Near homework

Cite this entry.

“Homework.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/homework. Accessed 17 Apr. 2024.

Kids Definition

Kids definition of homework, more from merriam-webster on homework.

Thesaurus: All synonyms and antonyms for homework

Nglish: Translation of homework for Spanish Speakers

Britannica English: Translation of homework for Arabic Speakers

Britannica.com: Encyclopedia article about homework

Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!

Play Quordle: Guess all four words in a limited number of tries. Each of your guesses must be a real 5-letter word.

Can you solve 4 words at once?

Word of the day, circumlocution.

See Definitions and Examples »

Get Word of the Day daily email!

Popular in Grammar & Usage

Your vs. you're: how to use them correctly, every letter is silent, sometimes: a-z list of examples, more commonly mispronounced words, how to use em dashes (—), en dashes (–) , and hyphens (-), absent letters that are heard anyway, popular in wordplay, a great big list of bread words, the words of the week - apr. 12, 10 scrabble words without any vowels, 12 more bird names that sound like insults (and sometimes are), 9 superb owl words, games & quizzes.

Play Blossom: Solve today's spelling word game by finding as many words as you can using just 7 letters. Longer words score more points.

Cambridge Dictionary +Plus

Meaning of homework in English

Your browser doesn't support HTML5 audio

The kids are busy with their homework.
My science teacher always sets a lot of homework.
"Have you got any homework tonight ?" "No."
I got A minus for my English homework.
For homework I want you to write an essay on endangered species .
academic year
access course
Advanced Placement
asynchronous
foundation course
immersion course
interdisciplinarity
on a course
open admissions
open classroom
work placement

homework | American Dictionary

Homework | business english, examples of homework, translations of homework.

Get a quick, free translation!

Word of the Day

balancing act

a difficult situation in which someone has to try to give equal amounts of importance, time, attention, etc. to two or more different things at the same time

Binding, nailing, and gluing: talking about fastening things together

Learn more with +Plus

Recent and Recommended {{#preferredDictionaries}} {{name}} {{/preferredDictionaries}}
Definitions Clear explanations of natural written and spoken English English Learner’s Dictionary Essential British English Essential American English
Grammar and thesaurus Usage explanations of natural written and spoken English Grammar Thesaurus
Pronunciation British and American pronunciations with audio English Pronunciation
English–Chinese (Simplified) Chinese (Simplified)–English
English–Chinese (Traditional) Chinese (Traditional)–English
English–Dutch Dutch–English
English–French French–English
English–German German–English
English–Indonesian Indonesian–English
English–Italian Italian–English
English–Japanese Japanese–English
English–Norwegian Norwegian–English
English–Polish Polish–English
English–Portuguese Portuguese–English
English–Spanish Spanish–English
English–Swedish Swedish–English
Dictionary +Plus Word Lists
English Noun
American Noun
do your homework
Translations
All translations

Add homework to one of your lists below, or create a new one.

Something went wrong.

There was a problem sending your report.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 8: Functions

About this unit.

A function is like a machine that takes an input and gives an output. Let's explore how we can graph, analyze, and create different types of functions.

Evaluating functions

What is a function? (Opens a modal)
Worked example: Evaluating functions from equation (Opens a modal)
Worked example: Evaluating functions from graph (Opens a modal)
Evaluating discrete functions (Opens a modal)
Worked example: evaluating expressions with function notation (Opens a modal)
Evaluate functions Get 3 of 4 questions to level up!
Evaluate functions from their graph Get 3 of 4 questions to level up!
Evaluate function expressions Get 3 of 4 questions to level up!

Inputs and outputs of a function

Worked example: matching an input to a function's output (equation) (Opens a modal)
Worked example: matching an input to a function's output (graph) (Opens a modal)
Worked example: two inputs with the same output (graph) (Opens a modal)
Function inputs & outputs: equation Get 3 of 4 questions to level up!
Function inputs & outputs: graph Get 3 of 4 questions to level up!

Functions and equations

Equations vs. functions (Opens a modal)
Obtaining a function from an equation (Opens a modal)
Function rules from equations Get 3 of 4 questions to level up!

Interpreting function notation

Function notation word problem: bank (Opens a modal)
Function notation word problem: beach (Opens a modal)
Function notation word problems Get 3 of 4 questions to level up!

Introduction to the domain and range of a function

Intervals and interval notation (Opens a modal)
What is the domain of a function? (Opens a modal)
What is the range of a function? (Opens a modal)
Worked example: domain and range from graph (Opens a modal)
Domain and range from graph Get 5 of 7 questions to level up!

Determining the domain of a function

Determining whether values are in domain of function (Opens a modal)
Examples finding the domain of functions (Opens a modal)
Worked example: determining domain word problem (real numbers) (Opens a modal)
Worked example: determining domain word problem (positive integers) (Opens a modal)
Worked example: determining domain word problem (all integers) (Opens a modal)
Identifying values in the domain Get 3 of 4 questions to level up!
Determine the domain of functions Get 3 of 4 questions to level up!
Function domain word problems Get 3 of 4 questions to level up!

Recognizing functions

Recognizing functions from graph (Opens a modal)
Does a vertical line represent a function? (Opens a modal)
Recognizing functions from table (Opens a modal)
Recognizing functions from verbal description (Opens a modal)
Recognizing functions from verbal description word problem (Opens a modal)
Recognize functions from graphs Get 3 of 4 questions to level up!
Recognize functions from tables Get 3 of 4 questions to level up!

Maximum and minimum points

Introduction to minimum and maximum points (Opens a modal)
Worked example: absolute and relative extrema (Opens a modal)
Relative maxima and minima Get 3 of 4 questions to level up!
Absolute maxima and minima Get 3 of 4 questions to level up!

Intervals where a function is positive, negative, increasing, or decreasing

Increasing, decreasing, positive or negative intervals (Opens a modal)
Worked example: positive & negative intervals (Opens a modal)
Positive and negative intervals Get 3 of 4 questions to level up!
Increasing and decreasing intervals Get 3 of 4 questions to level up!

Interpreting features of graphs

Graph interpretation word problem: temperature (Opens a modal)
Graph interpretation word problem: basketball (Opens a modal)
Creativity break: How can people get creative in algebra (Opens a modal)
Graph interpretation word problems Get 3 of 4 questions to level up!

Average rate of change

Introduction to average rate of change (Opens a modal)
Worked example: average rate of change from graph (Opens a modal)
Worked example: average rate of change from table (Opens a modal)
Average rate of change: graphs & tables Get 3 of 4 questions to level up!

Average rate of change word problems

Average rate of change word problem: table (Opens a modal)
Average rate of change word problem: graph (Opens a modal)
Average rate of change review (Opens a modal)
Average rate of change word problems Get 3 of 4 questions to level up!

Intro to inverse functions

Intro to inverse functions (Opens a modal)
Inputs & outputs of inverse functions (Opens a modal)
Graphing the inverse of a linear function (Opens a modal)
Finding inverse functions: linear (Opens a modal)
Functions: FAQ (Opens a modal)
Evaluate inverse functions Get 3 of 4 questions to level up!
Finding inverses of linear functions Get 3 of 4 questions to level up!

school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Download Page (PDF)
Download Full Book (PDF)
Periodic Table
Physics Constants
Scientific Calculator
Reference & Cite
Tools expand_more
Readability

selected template will load here

This action is not available.

2.2: Definition of the Derivative

Last updated
Save as PDF
Page ID 89714

Joel Feldman, Andrew Rechnitzer and Elyse Yeager

Joel Feldman, Andrew Rechnitzer and Elyse Yeager
University of British Columbia

We now define the “derivative” explicitly, based on the limiting slope ideas of the previous section. Then we see how to compute some simple derivatives.

Let us now generalise what we did in the last section so as to find “the slope of the curve $y=f(x)$ at $(x_0,y_0)$” for any smooth enough 1 function $f(x)\text{.}$

As before, let $(x_0,y_0)$ be any point on the curve $y=f(x)\text{.}$ So we must have $y_0=f(x_0)\text{.}$ Now let $(x_1,y_1)$ be any other point on the same curve. So $y_1=f(x_1)$ and $x_1\ne x_0\text{.}$ Think of $(x_1,y_1)$ as being pretty close to $(x_0,y_0)$ so that the difference

\begin{gather*} \Delta x=x_1-x_0 \end{gather*}

in $x$–coordinates is pretty small. In terms of this $\Delta x$ we have

\begin{gather*} x_1=x_0+\Delta x\qquad\text{and}\qquad y_1=f\big(x_0+\Delta x\big) \end{gather*}

We can construct a secant line through $(x_0,y_0)$ and $(x_1,y_1)$ just as we did for the parabola above. It has slope

\begin{gather*} \frac{y_1-y_0}{x_1-x_0}=\frac{f\big(x_0+\Delta x\big)-f(x_0)}{\Delta x} \end{gather*}

If $f(x)$ is reasonably smooth 2 , then as $x_1$ approaches $x_0\text{,}$ i.e. as $\Delta x$ approaches $0\text{,}$ we would expect the secant through $(x_0,y_0)$ and $(x_1,y_1)$ to approach the tangent line to the curve $y=f(x)$ at $(x_0,y_0)\text{,}$ just as happened in Figure 2.1.6. And more importantly, the slope of the secant through $(x_0,y_0)$ and $(x_1,y_1)$ should approach the slope of the tangent line to the curve $y=f(x)$ at $(x_0,y_0)\text{.}$

Thus we would expect 3 the slope of the tangent line to the curve $y=f(x)$ at $(x_0,y_0)$ to be

\begin{gather*} \lim_{\Delta x\rightarrow 0}\frac{f\big(x_0+\Delta x\big)-f(x_0)}{\Delta x} \end{gather*}

When we talk of the “slope of the curve” at a point, what we really mean is the slope of the tangent line to the curve at that point. So “the slope of the curve $y=f(x)$ at $(x_0,y_0)$” is also the limit 4 expressed in the above equation. The derivative of $f(x)$ at $x=x_0$ is also defined to be this limit. Which leads 5 us to the most important definition in this text:

Definition 2.2.1 Derivative at a point.

Let $a\in\mathbb{R}$ and let $f(x)$ be defined on an open interval 6 that contains $a\text{.}$

\begin{gather*} f'(a)=\lim_{h\rightarrow 0}\frac{f\big(a+h\big)-f(a)}{h} \end{gather*}

When the above limit exists, the function $f(x)$ is said to be differentiable at $x=a\text{.}$ When the limit does not exist, the function $f(x)$ is said to be not differentiable at $x=a\text{.}$

\begin{gather*} f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}. \end{gather*}

Lets now compute the derivatives of some very simple functions. This is our first step towards building up a toolbox for computing derivatives of complicated functions — this process will very much parallel what we did in Chapter 1 with limits. The two simplest functions we know are $f(x)=c$ and $g(x)=x\text{.}$

Example 2.2.2 Derivative of $f(x)=c$.

Let $a, c \in \mathbb{R}$ be a constants. Compute the derivative of the constant function $f(x) = c$ at $x=a\text{.}$

We compute the desired derivative by just substituting the function of interest into the formal definition of the derivative.

\begin{align*} f'(a) &= \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} && \text{(the definition)}\\ &= \lim_{h \to 0} \frac{c - c}{h} && \text{(substituted in the function)}\\ &= \lim_{h \to 0} 0 &&\text{(simplified things)}\\ &= 0 \end{align*}

That was easy! What about the next most complicated function — arguably it's this one:

Example 2.2.3 Derivative of $g(x)=x$.

Let $a\in \mathbb{R}$ and compute the derivative of $g(x) = x$ at $x=a\text{.}$

Again, we compute the derivative of $g$ by just substituting the function of interest into the formal definition of the derivative and then evaluating the resulting limit.

\begin{align*} g'(a) &= \lim_{h \to 0} \frac{g(a+h) - g(a)}{h} && \text{(the definition)}\\ &= \lim_{h \to 0} \frac{(a+h) - a}{h} && \text{(substituted in the function)}\\ &= \lim_{h \to 0} \frac{h}{h} && \text{(simplified things)}\\ &= \lim_{h \to 0} 1 && \text{(simplified a bit more)}\\ &= 1 \end{align*}

That was a little harder than the first example, but still quite straight forward — start with the definition and apply what we know about limits.

Thanks to these two examples, we have our first theorem about derivatives:

Theorem 2.2.4 Easiest derivatives.

Let $a,c \in \mathbb{R}$ and let $f(x) = c$ be the constant function and $g(x) = x\text{.}$ Then

\begin{align*} f'(a) &= 0\\ \end{align*}

To ratchet up the difficulty a little bit more, let us redo the example we have already done a few times $f(x)=x^2\text{.}$ To make it a little more interesting let's change the names of the function and the variable so that it is not exactly the same as Examples 2.1.2 and 2.1.5.

Example 2.2.5 Derivative of $h(t)=t^2$.

Compute the derivative of

\begin{align*} h(t) &= t^2 & \text{ at } t = a \end{align*}

This function isn't quite like the ones we saw earlier — it's a function of $t$ rather than $x\text{.}$ Recall that a function is a rule which assigns to each input value an output value. So far, we have usually called the input value $x\text{.}$ But this “$x$” is just a dummy variable representing a generic input value. There is nothing wrong with calling a generic input value $t$ instead. Indeed, from time to time you will see functions that are not written as formulas involving $x\text{,}$ but instead are written as formulas in $t$ (for example representing time — see Section 1.2), or $z$ (for example representing height), or other symbols.

\begin{align*} f'(a) &= \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} \end{align*}

\begin{align*} h'(a) &= \lim_{h \to 0} \frac{h(a+h)-h(a)}{h} \end{align*}

\begin{align*} h'(a) &= \lim_{\Delta t \to 0} \frac{h(a+\Delta t)-h(a)}{\Delta t} \end{align*}

\begin{align*} h'(a) &= \lim_{\Delta t \to 0} \frac{(a+\Delta t)^2-a^2}{\Delta t}\\ &= \lim_{\Delta t \to 0} \frac{a^2+2a\,\Delta t+\Delta t^2-a^2}{\Delta t} && \big(\text{just squared out $(a+\Delta t)^2$}\big)\\ &= \lim_{\Delta t \to 0} \frac{2a\,\Delta t+\Delta t^2}{\Delta t}\\ &= \lim_{\Delta t \to 0} (2a +\Delta t)\\ &= 2a \end{align*}

You should go back check that this is what we got in Example 2.1.5 — just some names have been changed.

An Important Point (and Some Notation)

Notice here that the answer we get depends on our choice of $a$ — if we want to know the derivative at $a=3$ we can just substitute $a=3$ into our answer $2a$ to get the slope is 6. If we want to know at $a=1$ (like at the end of Section 1.1) we substitute $a=1$ and get the slope is 2. The important thing here is that we can move from the derivative being computed at a specific point to the derivative being a function itself — input any value of $a$ and it returns the slope of the tangent line to the curve at the point $x=a\text{,}$ $y=h(a)\text{.}$ The variable $a$ is a dummy variable. We can rename $a$ to anything we want, like $x\text{,}$ for example. So we can replace every $a$ in

\begin{align*} h'(a)&=2a &\text{ by $x$, giving} && h'(x) &=2x \end{align*}

where all we have done is replaced the symbol $a$ by the symbol $x\text{.}$

We can do this more generally and tweak the derivative at a specific point $a$ to obtain the derivative as a function of $x\text{.}$ We replace

\begin{align*} f'(a) &= \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\\ \end{align*}

\begin{align*} f'(x) &= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \end{align*}

which gives us the following definition

Definition 2.2.6 Derivative as a function.

Let $f(x)$ be a function.

\begin{gather*} f'(x)=\lim_{h\rightarrow 0}\frac{f\big(x+h\big)-f(x)}{h} \end{gather*}

If the derivative $f'(x)$ exists for all $x \in (a,b)$ we say that $f$ is differentiable on $(a,b)\text{.}$
Note that we will sometimes be a little sloppy with our discussions and simply write “$f$ is differentiable” to mean “$f$ is differentiable on an interval we are interested in” or “$f$ is differentiable everywhere”.

Notice that we are no longer thinking of tangent lines, rather this is an operation we can do on a function. For example:

Example 2.2.7 The derivative of $f(x)=\tfrac{1}{x}$.

Let $f(x) = \frac{1}{x}$ and compute its derivative with respect to $x$ — think carefully about where the derivative exists.

\begin{align*} f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} && \text{(the definition)} \end{align*}

\begin{align*} f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} && \text{(the definition)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\left[\frac{1}{x+h}-\frac{1}{x}\right] && \text{(substituted in the function)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\ \frac{x-(x+h)}{x(x+h)} && \text{(wrote over a common denominator)}\\ &=\lim_{h\rightarrow 0}\frac{1}{h}\ \frac{-h}{x(x+h)} && \text{(started cleanup)}\\ &=\lim_{h\rightarrow 0} \frac{-1}{x(x+h)}\\ &=-\frac{1}{x^2} \end{align*}

Notice that the original function $f(x)=\frac{1}{x}$ was not defined at $x=0$ and the derivative is also not defined at $x=0\text{.}$ This does happen more generally — if $f(x)$ is not defined at a particular point $x=a\text{,}$ then the derivative will not exist at that point either.

So we now have two slightly different ideas of derivatives:

The derivative $f'(a)$ at a specific point $x=a\text{,}$ being the slope of the tangent line to the curve at $x=a\text{,}$ and
The derivative as a function, $f'(x)$ as defined in Definition 2.2.6.

Of course, if we have $f'(x)$ then we can always recover the derivative at a specific point by substituting $x=a\text{.}$

As we noted at the beginning of the chapter, the derivative was discovered independently by Newton and Leibniz in the late $17^{\rm th}$ century. Because their discoveries were independent, Newton and Leibniz did not have exactly the same notation. Stemming from this, and from the many different contexts in which derivatives are used, there are quite a few alternate notations for the derivative:

Definition 2.2.8.

The following notations are all used for “the derivative of $f(x)$ with respect to $x$”

\begin{gather*} f'(x) \qquad\frac{\mathrm{d} f}{\mathrm{d} x} \qquad\frac{\mathrm{d} f(x)}{\mathrm{d} x} \qquad \dot{f}(x) \qquad Df(x) \qquad D_x f(x), \end{gather*}

while the following notations are all used for “the derivative of $f(x)$ at $x=a$”

\begin{gather*} f'(a) \qquad\frac{\mathrm{d} f(a)}{\mathrm{d} x}\qquad \frac{\mathrm{d} f(x)}{\mathrm{d} x}\,\bigg|_{x=a} \qquad \dot{f}(a) \qquad Df(a) \qquad D_x f(a). \end{gather*}

Some things to note about these notations:

We will generally use the first three, but you should recognise them all. The notation $f'(a)$ is due to Lagrange, while the notation $\frac{\mathrm{d} f(a)}{\mathrm{d} x}$ is due to Leibniz. They are both very useful. Neither can be considered “better”.
Leibniz notation writes the derivative as a “fraction” — however it is definitely not a fraction and should not be thought of in that way. It is just shorthand, which is read as “the derivative of $f$ with respect to $x$”.
You read $f'(x)$ as “$f$–prime of $x$”, and $\frac{\mathrm{d} f}{\mathrm{d} x}$ as “dee–$f$–dee–$x$”, and $\frac{\mathrm{d} f(x)}{\mathrm{d} x}$ as “dee-by-dee–$x$ of $f$”.
Similarly you read $\frac{\mathrm{d} f(a)}{\mathrm{d} x}$ as “dee–$f$–dee–$x$ at $a$”, and $\frac{\mathrm{d} f(x)}{\mathrm{d} x}\,\bigg|_{x=a}$ as “dee-by-dee-$x$ of $f$ of $x$ at $x$ equals $a$”.
The notation $\dot f$ is due to Newton. In physics, it is common to use $\dot f(t)$ to denote the derivative of $f$ with respect to time.

Back to Computing Some Derivatives

At this point we could try to start working out how derivatives interact with arithmetic and make an “Arithmetic of derivatives” theorem just like the one we saw for limits (Theorem 1.4.3). We will get there shortly, but before that it is important that we become more comfortable with computing derivatives using limits and then understanding what the derivative actually means. So — more examples.

Example 2.2.9 $\frac{\mathrm{d} }{\mathrm{d} x}\sqrt{x}$.

Compute the derivative, $f'(a)\text{,}$ of the function $f(x)=\sqrt{x}$ at the point $x=a$ for any $a \gt 0\text{.}$

\begin{align*} f'(a) &=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a} =\lim_{x\rightarrow a}\frac{\sqrt{x}-\sqrt{a}}{x-a} \end{align*}

As $x$ tends to $a\text{,}$ the numerator and denominator both tend to zero. But $\tfrac{0}{0}$ is not defined. So to get a well defined limit we need to exhibit a cancellation between the numerator and denominator — just as we saw in Examples 1.4.12 and 1.4.17. Now there are two equivalent ways to proceed from here, both based on a similar “trick”.

\begin{align*} &\frac{\sqrt{x}-\sqrt{a}}{x-a}\\ &= \frac{\sqrt{x}-\sqrt{a}}{x-a} \times \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}} && \Big(\text{multiplication by $1=\frac{\text{conjugate}}{\text{conjugate}}$}\Big)\\ &=\frac{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})} {(x-a)(\sqrt{x}+\sqrt{a})}\\ &= \frac{x-a}{(x-a)(\sqrt{x}+\sqrt{a})} && \big(\text{since $(A-B)(A+B) = A^2-B^2$)}\,\big)\\ &= \frac{1}{\sqrt{x}+\sqrt{a}} \end{align*}

\begin{align*} x - a &= (\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a}) \end{align*}

\begin{align*} \frac{\sqrt{x}-\sqrt{a}}{x-a} &=\frac{\sqrt{x}-\sqrt{a}}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})} & \text{(now cancel common factors)}\\ &=\frac{1}{(\sqrt{x}+\sqrt{a})} \end{align*}

\begin{align*} f'(a) &=\lim_{x\rightarrow a}\frac{\sqrt{x}-\sqrt{a}}{x-a}\\ & =\lim_{x\rightarrow a}\frac{1}{\sqrt{x}+\sqrt{a}}\\ & =\frac{1}{2\sqrt{a}} \end{align*}

If we draw a careful picture of $\sqrt{x}$ around $x=0$ we can see why this has to be the case. The figure below shows three different tangent lines to the graph of $y=f(x)=\sqrt{x}\text{.}$ As the point of tangency moves closer and closer to the origin, the tangent line gets steeper and steeper. The slope of the tangent line at $\big(a,\sqrt{a}\big)$ blows up as $a\to 0\text{.}$

Example 2.2.10 $\frac{\mathrm{d} }{\mathrm{d} x}\left\{ |x| \right\}$.

Compute the derivative, $f'(a)\text{,}$ of the function $f(x)=|x|$ at the point $x=a\text{.}$

\begin{align*} |x| &= \begin{cases} -x & \text{ if } x \lt 0\\ 0 & \text{ if } x=0\\ x & \text{ if }x \gt 0. \end{cases} \end{align*}

It is definitely not just “chop off the minus sign”.

This breaks our computation of the derivative into 3 cases depending on whether $x$ is positive, negative or zero.

\begin{align*}\frac{\mathrm{d} f}{\mathrm{d} x} &= \lim_{h\to0} \frac{f(x+h)-f(x)}{h}\\ &= \lim_{h\to0} \frac{|x+h|-|x|}{h}\\ \\ \end{align*}

Since $x \gt 0$ and we are interested in the behaviour of this function as $h \to 0$ we can assume $h$ is much smaller than $x\text{.}$ This means $x+h \gt 0$ and so $|x+h|=x+h\text{.}$

\begin{align*} \frac{\mathrm{d} f}{\mathrm{d} x} &= \lim_{h\to0} \frac{f(x+h)-f(x)}{h}\\ &= \lim_{h\to0} \frac{|x+h|-|x|}{h}\\ \\ \end{align*}

Since $x \lt 0$ and we are interested in the behaviour of this function as $h \to 0$ we can assume $h$ is much smaller than $x\text{.}$ This means $x+h \lt 0$ and so $|x+h|=-(x+h)\text{.}$

\begin{align*} f'(0) &= \lim_{h\to0} \frac{f(0+h)-f(0)}{h}\\ &= \lim_{h\to0} \frac{|0+h|-|0|}{h}\\ &= \lim_{h\to0} \frac{|h|}{h} \end{align*}

\begin{align*} \lim_{h \to 0^+} \frac{|h|}{h} &= \lim_{h \to 0^+} \frac{h}{h} &\text{since } h \gt 0, |h|=h\\ &= 1\\ \end{align*}

Whereas, the limit from below is:

In summary:

\begin{align*} \frac{\mathrm{d}}{\mathrm{d} x} |x| &= \begin{cases} -1 & \text{if } x \lt 0 \\ DNE & \text{if } x=0 \\ 1 & \text{if } x \gt 0 \end{cases} \end{align*}

Where is the Derivative Undefined?

According to Definition 2.2.1, the derivative $f'(a)$ exists precisely when the limit $\lim\limits_{x\rightarrow a} \frac{f(x)-f(a)}{x-a}$ exists. That limit is also the slope of the tangent line to the curve $y=f(x)$ at $x=a\text{.}$ That limit does not exist when the curve $y=f(x)$ does not have a tangent line at $x=a$ or when the curve does have a tangent line, but the tangent line has infinite slope. We have already seen some examples of this.

In Example 2.2.7, we considered the function $f(x)=\frac{1}{x}\text{.}$ This function “blows up” (i.e. becomes infinite) at $x=0\text{.}$ It does not have a tangent line at $x=0$ and its derivative does not exist at $x=0\text{.}$
In Example 2.2.10, we considered the function $f(x)=|x|\text{.}$ This function does not have a tangent line at $x=0\text{,}$ because there is a sharp corner in the graph of $y=|x|$ at $x=0\text{.}$ (Look at the graph in Example 2.2.10.) So the derivative of $f(x)=|x|$ does not exist at $x=0\text{.}$

Here are a few more examples.

Example 2.2.11 Derivative at a discontinuity.

Visually, the function

$H(x) = \begin{cases} 0 & \text{if }x \le 0 \\ 1 & \text{if }x \gt 0 \end{cases}$

does not have a tangent line at $(0,0)\text{.}$ Not surprisingly, when $a=0$ and $h$ tends to $0$ with $h \gt 0\text{,}$

\begin{gather*} \frac{H(a+h)-H(a)}{h} =\frac{H(h)-H(0)}{h} =\frac{1}{h} \end{gather*}

blows up. The same sort of computation shows that $f'(a)$ cannot possibly exist whenever the function $f$ is not continuous at $a\text{.}$ We will formalize, and prove, this statement in Theorem 2.2.14, below.

Example 2.2.12 $\frac{\mathrm{d} }{\mathrm{d} x}x^{1/3}$.

Visually, it looks like the function $f(x) = x^{1/3}\text{,}$ sketched below, (this might be a good point to recall that cube roots of negative numbers are negative — for example, since $(-1)^3=-1\text{,}$ the cube root of $-1$ is $-1$),

has the $y$–axis as its tangent line at $(0,0)\text{.}$ So we would expect that $f'(0)$ does not exist. Let's check. With $a=0\text{,}$

\begin{align*} f'(a)&= \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h} =\lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h} =\lim_{h\rightarrow 0}\frac{h^{1/3}}{h}\\ &=\lim_{h\rightarrow 0}\frac{1}{h^{2/3}} =DNE \end{align*}

as expected.

Example 2.2.13 $\frac{\mathrm{d} }{\mathrm{d} x}\sqrt{\left | x \right |}$.

We have already considered the derivative of the function $\sqrt{x}$ in Example 2.2.9. We'll now look at the function $f(x) = \sqrt{|x|}\text{.}$ Recall, from Example 2.2.10, the definition of $|x|\text{.}$

When $x \gt 0\text{,}$ we have $|x|=x$ and $f(x)$ is identical to $\sqrt{x}\text{.}$ When $x \lt 0\text{,}$ we have $|x|=-x$ and $f(x)=\sqrt{-x}\text{.}$ So to graph $y=\sqrt{|x|}$ when $x \lt 0\text{,}$ you just have to graph $y=\sqrt{x}$ for $x \gt 0$ and then send $x\rightarrow -x$ — i.e. reflect the graph in the $y$–axis. Here is the graph.

The pointy thing at the origin is called a cusp. The graph of $y=f(x)$ does not have a tangent line at $(0,0)$ and, correspondingly, $f'(0)$ does not exist because

\begin{gather*} \lim_{h\rightarrow 0^+}\frac{f(h)-f(0)}{h} =\lim_{h\rightarrow 0^+}\frac{\sqrt{|h|}}{h} =\lim_{h\rightarrow 0^+}\frac{1}{\sqrt{h}} =DNE \end{gather*}

Theorem 2.2.14.

If the function $f(x)$ is differentiable at $x=a\text{,}$ then $f(x)$ is also continuous at $x=a\text{.}$

The function $f(x)$ is continuous at $x=a$ if and only if the limit of

\begin{gather*} f(a+h) - f(a) = \frac{f(a+h)-f(a)}{h}\ h \end{gather*}

as $h\rightarrow 0$ exists and is zero. But if $f(x)$ is differentiable at $x=a\text{,}$ then, as $h\rightarrow 0\text{,}$ the first factor, $\frac{f(a+h)-f(a)}{h}$ converges to $f'(a)$ and the second factor, $h\text{,}$ converges to zero. So the product provision of our arithmetic of limits Theorem 1.4.3 implies that the product $\frac{f(a+h)-f(a)}{h}\ h$ converges to $f'(a)\cdot 0=0$ too.

Notice that while this theorem is useful as stated, it is (arguably) more often applied in its contrapositive 7 form:

Theorem 2.2.15 The contrapositive of Theorem 2.2.14.

If $f(x)$ is not continuous at $x=a$ then it is not differentiable at $x=a\text{.}$

As the above examples illustrate, this statement does not tell us what happens if $f$ is continuous at $x=a$ — we have to think!

Exercise $\PageIndex{1}$

The function $f(x)$ is shown. Select all options below that describe its derivative, $\frac{\mathrm{d} f}{\mathrm{d} x}\text{:}$

(a) constant
(b) increasing
(c) decreasing
(d) always positive
(e) always negative

Exercise $\PageIndex{2}$

Exercise $\pageindex{3}$, exercise $\pageindex{4}$(✳).

State, in terms of a limit, what it means for $f(x) = x^3$ to be differentiable at $x = 0\text{.}$

Exercise $\PageIndex{5}$

For which values of $x$ does $f'(x)$ not exist?

Exercise $\PageIndex{6}$

Suppose $f(x)$ is a function defined at $x=a$ with

\[ \lim_{h \to 0^-}\frac{f(a+h)-f(a)}{h}=\lim_{h \to 0^+}\frac{f(a+h)-f(a)}{h}=1. \nonumber \]

True or false: $f'(a)=1\text{.}$

Exercise $\PageIndex{7}$

\[ \lim_{x \to a^-}f'(x)=\lim_{x \to a^+}f'(x)=1. \nonumber \]

Exercise $\PageIndex{8}$

Suppose $s(t)$ is a function, with $t$ measured in seconds, and $s$ measured in metres. What are the units of $s'(t)\text{?}$

Exercise $\PageIndex{9}$

Use the definition of the derivative to find the equation of the tangent line to the curve $y(x)=x^3+5$ at the point $(1,6)\text{.}$

Exercise $\PageIndex{10}$

Use the definition of the derivative to find the derivative of $f(x)=\frac{1}{x}\text{.}$

Exercise $\PageIndex{11}$(✳)

Let $f(x) = x|x|\text{.}$ Using the definition of the derivative, show that $f(x)$ is differentiable at $x = 0\text{.}$

Exercise $\PageIndex{12}$(✳)

Use the definition of the derivative to compute the derivative of the function $f(x)=\frac{2}{x+1}\text{.}$

Exercise $\PageIndex{13}$(✳)

Use the definition of the derivative to compute the derivative of the function $f(x)=\frac{1}{x^2+3}\text{.}$

Exercise $\PageIndex{14}$

Use the definition of the derivative to find the slope of the tangent line to the curve $f(x)=x\log_{10}(2x+10)$ at the point $x=0\text{.}$

Exercise $\PageIndex{15}$(✳)

Compute the derivative of $f(x)=\frac{1}{x^2}$ directly from the definition.

Exercise $\PageIndex{16}$(✳)

Find the values of the constants $a$ and $b$ for which

\begin{align*} f(x) = \left\{ \begin{array}{lc} x^2 & x\le 2\\ ax + b & x \gt 2 \end{array}\right. \end{align*}

is differentiable everywhere.

Remark: In the text, you have already learned the derivatives of $x^2$ and $ax+b\text{.}$ In this question, you are only asked to find the values of $a$ and $b$—not to justify how you got them—so you don't have to use the definition of the derivative. However, on an exam, you might be asked to justify your answer, in which case you would show how to differentiate the two branches of $f(x)$ using the definition of a derivative.

Exercise $\PageIndex{17}$(✳)

Use the definition of the derivative to compute $f'(x)$ if $f(x) = \sqrt{1 + x}\text{.}$ Where does $f'(x)$ exist?

Exercise $\PageIndex{18}$

Use the definition of the derivative to find the velocity of an object whose position is given by the function $s(t)=t^4-t^2\text{.}$

Exercise $\PageIndex{19}$(✳)

Determine whether the derivative of following function exists at $x=0\text{.}$

\begin{align*} f(x) &=\begin{cases} x \cos x & \text{ if } x\ge 0\\ \sqrt{x^2+x^4} & \text{ if } x \lt 0 \end{cases} \end{align*}

You must justify your answer using the definition of a derivative.

Exercise $\PageIndex{20}$(✳)

Determine whether the derivative of the following function exists at $x=0$

\begin{align*} f(x) &=\begin{cases} x \cos x & \text{ if } x\le 0\\ \sqrt{1+x}-1 & \text{ if } x \gt 0 \end{cases} \end{align*}

Exercise $\PageIndex{21}$(✳)

\begin{align*} f(x) &=\begin{cases} x^3-7x^2 & \text{ if } x\le 0\\ x^3 \cos\left(\frac{1}{x}\right) & \text{ if } x \gt 0 \end{cases} \end{align*}

Exercise $\PageIndex{22}$(✳)

Determine whether the derivative of the following function exists at $x=1$

\begin{align*} f(x) &=\begin{cases} 4x^2-8x+4 & \text{ if } x\le 1\\ (x-1)^2\sin\left(\dfrac{1}{x-1}\right) & \text{ if } x \gt 1 \end{cases} \end{align*}

Exercise $\PageIndex{23}$

Sketch a function $f(x)$ with $f'(0)=-1$ that takes the following values:

Remark: you can't always guess the behaviour of a function from its points, even if the points seem to be making a clear pattern.

Exercise $\PageIndex{24}$

Let $p(x)=f(x)+g(x)\text{,}$ for some functions $f$ and $g$ whose derivatives exist. Use limit laws and the definition of a derivative to show that $p'(x)=f'(x)+g'(x)\text{.}$

Remark: this is called the sum rule, and we'll learn more about it in Lemma 2.4.1.

Exercise $\PageIndex{25}$

Let $f(x)=2x\text{,}$ $g(x)=x\text{,}$ and $p(x)=f(x) \cdot g(x)\text{.}$

Find $f'(x)$ and $g'(x)\text{.}$
Find $p'(x)\text{.}$
Is $p'(x)=f'(x) \cdot g'(x)\text{?}$

In Theorem 2.4.3, you'll learn a rule for calculating the derivative of a product of two functions.

Exercise $\PageIndex{26}$(✳)

There are two distinct straight lines that pass through the point $(1,-3)$ and are tangent to the curve $y = x^2\text{.}$ Find equations for these two lines.

Remark: the point $(1,-3)$ does not lie on the curve $y=x^2\text{.}$

Exercise $\PageIndex{27}$(✳)

For which values of $a$ is the function

\[ f(x) =\left\{\begin{array}{ll} 0 & x\le 0\\ x^a \sin\frac{1}{x} & x \gt 0\end{array}\right. \nonumber \]

differentiable at 0?

IMAGES

Benefits of homework in mathematics
7 Types of Homework for Students (2023) (2023)
Mathematics Homework Book Minzc 34
Definition of mathematics
What is Mathematics? Definition, Branches, Books and Mathematicians
Pros And Cons Of Doing Homework

VIDEO

finally I know what's really homework means
The meaning of school math and homework
COMBUSTION
What is Mathematics ?
Math Homework Helpers
PERIOD

COMMENTS

PDF Role of Homework in Mathematics
This study investigates some students' and teachers' beliefs about the role of homework in mathematics in Sweden. By interviewing eight adolescent students and two experienced mathematics teachers the study discusses the role of homework in relation to learning, teaching, achievements, and relationship between students, teachers and parents.
Mathematics homework and the potential compounding of educational
Mathematics homework is the specific focus of this paper. As an extension of the classroom, mathematics homework is shaped by the practices and relationships relevant to the teaching context (Kemmis and Grootenboer Citation 2008). Yet, how children's home lives, social experiences, and pre-existing skills interact with broader education ...
Math Homework: What to Expect and Why IT Is Important
Math Homework. Math homework is any task assigned to students to complete outside of their math class, and is created to help students prepare to learn new mathematical concepts, practice ones that have already been introduced, and explore other math skills. These out-of-class assignments are help to reinforce the lessons a child is introduced ...
Full article: The Creation and Implementation of Effective Homework
1. EFFECTIVE HOMEWORK PRACTICES. This issue of PRIMUS is the second of a two-part special issue on The Creation and Implementation of Effective Homework Assignments. Part 1 of the special issue focused on the creation of effective homework and featured papers that discussed elements of effective homework design and presented innovative homework systems targeting specific learning goals.
Online Mathematics Homework Increases Student Achievement
The purpose of mathematics homework is typically to provide practice for the student. Literature reviews and meta-analyses show generally positive or neutral effects for homework on learning (Cooper, Robinson, & Patall, 2006; Maltese, Robert, & Fan, 2012).Effects due to homework are more positive in middle and high school than elementary school (reflecting greater student maturity) and ...
PDF Maximizing the Benefits of Mathematics Homework: A Professional
MAXIMIZING THE BENEFITS OF MATHEMATICS HOMEWORK 5 . Chapter 1: Introduction There is currently a debate over the significance of mathematics homework. Parents and teachers seem to disagree as to whether or not mathematics homework plays a significant and positive role in a student's academic success.
Towards a theory of mathematics homework as a social practice
Abstract and Figures. This article presents a theoretical conceptualization of mathematics homework as a social practice. Rather than considering homework as a task or an artifact, this approach ...
How mathematicians assign homework problems in abstract ...
They are (1) knowing and recalling axioms and definitions, (2) developing an arsenal of examples, (3) developing new problem approaches, (4) remediating misconceptions, (5) making connections to prior and future material, and (6) valuing reading notes or text. ... This trend is present in research on mathematics homework at the university level ...
6.3: Homework
First, state what the problem means, and then explain and show each step you need to take to find the answer. Each problem requires 3 or 4 steps. a. 6 - 8. HW #11. For each of the following sets, determine if the set is closed under the operation given. Provide a counterexample if it is not closed. a.
Towards a theory of mathematics homework as a social practice
In Wenger's social theory of learning, individuals "negotiate" the meaning of their work in and across communities of practice (Wenger, 1998, pp. 53-55). In the practice of mathematics homework, students participate in academic and social communities, such as mathematics classes, schools, teams, clubs, and families.
Does homework design matter? The role of homework's purpose in student
This study used a randomized pretest-posttest clustered design to examine the effect of 3 homework purposes (i.e., practice, preparation, and extension) on 6th graders' mathematics achievement and how this relationship was modulated by the amount of completed homework.A total of 27 mathematics teachers and their 638 students participated in this study.
"Homework Should Be…but We Do Not Live in an Ideal World": Mathematics
Despite the extended use of this homework purpose by teachers, a recent study conducted with mathematics teachers found that homework with the purpose of practicing the material covered in class did not impact significantly the academic achievement of 6th-grade students; however, homework designed with the purpose of solving problems did ...
"Homework Feedback Is…": Elementary and Middle School ...
68) found that "class discussion on homework," and grading and commenting on homework were the practices most frequently used by high school teachers (i.e., English, mathematics, science, and social science) to monitor students' completion of homework. Focusing on mathematics, Kaur (2011) explored the nature of homework tasks assigned by ...
mathematics
Mathematics, or math, is often defined as the study of quantity, magnitude, and relations of numbers or symbols. It embraces the subjects of arithmetic, geometry, algebra, calculus, probability, statistics, and many other special areas of research. There are two major divisions of mathematics: pure and applied.
Homework Definition & Meaning
How to use homework in a sentence. piecework done at home for pay; an assignment given to a student to be completed outside the regular class period… See the full definition
THE CASE FOR (QUALITY) HOMEWORK: WHY IT IMPROVES LEARNING, AND ...
Noting that PISA studies have consistently found that spending more time on math homework strongly correlates with higher academic achievement, the report's authors suggest that the homework disparity may reflect lower teacher expectations for low-income students. If so, this is truly unfortunate. In and of itself, low socioeconomic status is ...
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a ...
What is a homework question?
1. You are right. However a person learning mathematics (he/she may be a school teacher or a mathematics professor) can be his own teacher (that's why the word self-teaching was invented). In this sense, if a person asks a mathematical question to himself to learn something, it can be seen as sort of homework.
PROTOCOL: The relationship between homework time and academic
1.1. Description of the condition. Homework is defined as "any task assigned by schoolteachers intended for students to carry out during non‐school hours" (Cooper, 1989).This definition explicitly excludes (a) in‐school guided study; (b) home study courses delivered through the mail, television, audio or videocassette, or the internet; and (c) extracurricular activities such as sports ...
HOMEWORK
HOMEWORK definition: 1. work that teachers give their students to do at home: 2. work that teachers give their students…. Learn more.
Functions
Unit test. Level up on all the skills in this unit and collect up to 2,200 Mastery points! A function is like a machine that takes an input and gives an output. Let's explore how we can graph, analyze, and create different types of functions.
2.15: Homework- Examples of the Definition of the Derivative
This page titled 2.15: Homework- Examples of the Definition of the Derivative is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
2.2: Definition of the Derivative
The derivative f ′ (a) at a specific point x = a, being the slope of the tangent line to the curve at x = a, and. The derivative as a function, f ′ (x) as defined in Definition 2.2.6. Of course, if we have f ′ (x) then we can always recover the derivative at a specific point by substituting x = a.

Math Homework: What to Expect and Why IT Is Important

Math Homework

Too Much Math Homework?

Other Articles You May Be Interested In

We Found 7 Tutors You Might Be Interested In

Kaplan Kids

Sylvan Learning

Tutor Doctor

Find the Perfect Tutor

Want to Help Your Child Learn?

Towards a theory of mathematics homework as a social practice

Cite this article

Access this article

Similar content being viewed by others

Movement in the Mathematics Classroom

Instruction and Construction of Mathematics at Home: An Exploratory Study

Acknowledgments

Author information

Corresponding author

Rights and permissions

About this article

Share this article

“Homework Should Be…but We Do Not Live in an Ideal World”: Mathematics Teachers’ Perspectives on Quality Homework and on Homework Assigned in Elementary and Middle Schools

Jennifer Cunha

Associated Data

Introduction

Research Background on Homework Characteristics

The Present Study

Materials and Methods

Participants

Data Collection

Data Analysis

Initial Data Screening

How do Elementary and Middle School Teachers Perceive Quality Homework?

How do Elementary and Middle School Teachers Describe the Homework Tasks They Typically Assign to Students?

Discussion and Implications for Practice and Research

Strengths and Limitations of the Study

Ethics Statement

Author Contributions

Conflict of Interest Statement

Acknowledgments

Supplementary Material

ORIGINAL RESEARCH article

Introduction

Teachers’ Role on the Homework Feedback Process

Students’ Role on the Homework Feedback Process

The Benefits of Homework Feedback

The Present Study

Materials and Methods

Participants in Focus Groups

Participants in Classroom Observations

Data Collection

Focus Group Discussions

Classroom Observations

Data Analysis

Initial Data Screening

Definitions of Homework Feedback

Purposes of Homework Feedback

Types of Homework Feedback Practices

Perceived Impact of Homework Feedback

Relationships between Teachers’ Conceptions of Homework Feedback

Practical Implications

Strengths, Limitations, and Future Research

Ethics Statement

Author Contributions

Conflict of Interest Statement

Acknowledgments

Definition of homework

Word History

Dictionary Entries Near homework

Kids Definition

Can you solve 4 words at once?

Popular in Grammar & Usage

Meaning of homework in English

homework | American Dictionary

Learn more with +Plus

Unit 8: Functions

Evaluating functions

Inputs and outputs of a function

Functions and equations