Skip to main content
SpringerLink
Log in

Development of a Sustainable Place Value Understanding

  • Chapter
  • First Online:
Book cover International Handbook of Mathematical Learning Difficulties

Abstract

A resilient understanding of the decimal place value system is a crucial element of primary mathematical education that has a deep impact on further arithmetical development. Place value understanding has been identified as a good predictor of math performances as well as math difficulties. Difficulties in understanding the place value system affect children in different grades in countries all over the world.

Prior and recent research has been focused on ordinal counting schemes and computing strategies. Much attention has been given to transcoding procedures that transfer numerals, number words, and magnitude representations to each other. However, the cardinal aspect of bundling and unbundling has not been taken into account. In this chapter we aim to present a sequence of levels of concepts that describe children’s development of place value understanding. The presented sequence integrates aspects of counting and bundling—in particular, regarding nonstandard partitions. Bundling tasks seem to be more crucial for place value understanding than transcoding tasks. Our own recent empirical studies underpin the hierarchy of the sequence levels.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aunio, P., & Räsänen, P. (2016). Core numerical skills for learning mathematics in children aged five to eight years – a working model for educators. European Early Childhood Education Research, 24(5), 684–704.

    Article  Google Scholar 

  • Byrge, L., Smith, L., & Mix, K. (2014). Beginnings of place value: How preschoolers write three-digit numbers. Child Development, 85, 437–443.

    Article  Google Scholar 

  • Cawley, J. F., Parmar, R. S., Lucas-Fusco, L. M., Kilian, J. D., & Foley, T. E. (2007). Place value and mathematics for students with mild disabilities: Data and suggested practices. Learning Disabilities: A Contemporary Journal, 3(1), 21–39.

    Google Scholar 

  • Chan, B. M. Y., & Ho, C. S. H. (2010). The cognitive profile of Chinese children with mathematics difficulties. Journal of Experimental Child Psychology, 107, 260–279.

    Article  Google Scholar 

  • Chan, W. W. L., Au, T. K., & Tang, J. (2014). Strategic counting: A novel assessment of place-value understanding. Learning and Instruction, 29, 78–94.

    Article  Google Scholar 

  • Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten. Focus on Learning Problems in Mathematics, 10(3), 1–28.

    Google Scholar 

  • Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–40.

    Article  Google Scholar 

  • Dehaene, S. (2011). The number sense: How the mind creates mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Desoete, A. (2015). Cognitive predictors of mathematical abilities and disabilities. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of mathematical cognition (pp. 915–932). Oxford: 2 Medicine UK.

    Google Scholar 

  • Desoete, A., Ceulemans, A., Roeyers, H., & Huylebroeck, A. (2009). Subitizing or counting as possible screening variables for learning disabilities in mathematics education or learning? Educational Research Review, 4, 55–66.

    Article  Google Scholar 

  • Dowker, A., & Morris, P. (2015). Interventions for children with difficulties in learning mathematics. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 256–264). London: Routledge.

    Google Scholar 

  • Dowker, A., & Roberts, M. (2015). Does the transparency of the counting system affect children’s numerical abilities? Frontiers in Psychology, 6, 945.

    Article  Google Scholar 

  • Dunne, T., Long, C., Craig, T., & Venter, E. (2012). Meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: The potential of Rasch measurement theory. Pythagoras, 33(3), 16p.

    Google Scholar 

  • Fritz, A., Ehlert, A., & Balzer, L. (2013). Development of mathematical concepts as basis for an elaborated mathematical understanding. South African Journal for Childhood Education, 3(1), 38–67.

    Google Scholar 

  • Fritz, A., & Ricken, G. (2008). Rechenschwäche. Basel: Reinhardt.

    Book  Google Scholar 

  • Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21(3), 180–206.

    Article  Google Scholar 

  • Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., et al. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162.

    Article  Google Scholar 

  • Gebhardt, M., Zehner, F., & Hessels, M. G. P. (2012). Basic arithmetical skills of students with learning disabilities in the secondary special schools. An exploratory study covering fifth to ninth grade. Frontline Learning Research, 3, 50–63.

    Google Scholar 

  • Gervasoni, A., & Sullivan, P. (2007). Assessing and teaching children who have difficulty learning arithmetic. Educational & Child Psychology, 24(2), 40–53.

    Google Scholar 

  • Hart, K. (2009). Why do we expect so much? In J. Novotná, & H. Moraova (Eds.), SEMT 2009. International symposium elementary maths teaching. August 23–28, 2009. Proceedings: The development of mathematical understanding (pp. 24–31). Prague, Czech Republic: Charles University.

    Google Scholar 

  • Herzog, M., Ehlert, A., & Fritz, A. (2017). A competency model of place value understanding in South African primary school pupils. African Journal of Research in Mathematics, Science and Technology Education, 21(1), 37–48.

    Article  Google Scholar 

  • Herzog, M., Ehlert, A. & Fritz, A. (in prep). The development of place value concepts across primary school.

    Google Scholar 

  • Herzog, M., Fritz, A., & Ehlert, A. (2017). Entwicklung eines tragfähigen Stellenwertverständnisses [Development of a resilient place value understanding]. In A. Fritz & S. Schmidt (Eds.), Handbuch Rechenschwäche (pp. 266–286). Basel: Beltz.

    Google Scholar 

  • Ho, S. H., & Cheng, F. S.-F. (1997). Training in place-value concepts improves children’s addition skills. Contemporary Educational Psychology, 22, 495–506.

    Article  Google Scholar 

  • Ifrah, G. (1998). The universal history of numbers. From prehistory to the invention of the computer. London: Harville Press.

    Google Scholar 

  • Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1(2), 75–86.

    Article  Google Scholar 

  • Ladel, S., & Kortenkamp, U. (2016). Development of a flexible understanding of place value. In T. Meaney, O. Helenius, M.-L. Johansson, T. Lange, & A. Wernberg (Eds.), Mathematics education in the early years – results from the POEM2 conference (pp. 289–307). New York: Springer.

    Chapter  Google Scholar 

  • Lonnemann, J., & Hasselhorn, M. (2018). Assessing mathematical competence and performance: Quality characteristics, approaches, and research trends. In A. Fritz-Stratmann, V. G. Haase, & P. Räsänen (Eds.), The international handbook of math learning difficulties: from the lab to the classroom. São Paulo, Brazil: Springer Brazil.

    Google Scholar 

  • MacDonald, A. (2008). But what about the Oneths? A year 7 Student’s misconception about decimal place value. Australian Mathematics Teacher, 64(4), 12–15.

    Google Scholar 

  • Mark, W., & Dowker, A. (2015). Linguistic influence on mathematical development is specific rather than pervasive: Revisiting the Chinese number advantage in Chinese and English children. Frontiers in Psychology, 6, 203.

    Article  Google Scholar 

  • Miller, K. F., Smith, C. M., Zhu, J., & Zhang, H. (1995). Preschool origins of cross-national differences in mathematical competence: The role of number-naming systems. Psychological Science, 6(1), 56–60.

    Article  Google Scholar 

  • Miura, I., Kim, C., Chang, C.-M., & Okamoto, Y. (1988). Effects of language characteristics on children’s cognitive representation of number: Cross-national comparisons. Child Development, 59(6), 1145–1150.

    Article  Google Scholar 

  • Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H.-C. (2011). Early place-value understanding as a precursor for later arithmetic performance – A longitudinal study on numerical development. Research in Developmental Disabilities, 32(5), 1837–1851.

    Article  Google Scholar 

  • Nuerk, H.-C., Moeller, K., & Willmes, K. (2015). Multi-digit number processing: Overview, conceptual clarifications, and language influences. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of mathematical cognition (pp. 106–139). Oxford: 2 Medicine UK.

    Google Scholar 

  • Nührenbörger, M., & Steinbring, H. (2008). Manipulatives as tools in teacher education. In D. Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education Vol. 2. Tools and processes in mathematics teacher education (pp. 157–182). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Okamoto, Y. (2015). Mathematics learning in the USA and Japan. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of mathematical cognition (pp. 415–429). Oxford: 2 Medicine UK.

    Google Scholar 

  • Pixner, S., Zuber, J., Heřmanová, V., Kaufmann, L., Nuerk, H.-C., & Moeller, K. (2011). One language, two number-word systems and many problems: Numerical cognition in the Czech language. Research in Developmental Disabilities, 32, 2683–2689.

    Article  Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 9, 346–362.

    Article  Google Scholar 

  • Ross, S. H. (1989). Parts, wholes and place value: A developmental view. The Arithmetic Teacher, 36(6), 47–51.

    Article  Google Scholar 

  • Schulz, A. (2014). Fachdidaktisches Wissen von Grundschullehrkräften [Content knowledge of primary school teachers]. Wiesbaden, Germany: Springer.

    Google Scholar 

  • Van de Walle, J. A., Karp, K., & Bay-Williams, J. M. (2004). Elementary and middle school mathematics. Teaching developmentally. Boston: Pearson.

    Google Scholar 

  • Zaslavsky, C. (1999). Africa counts. Number and pattern in African cultures. Chicago: Francis Hills.

    Google Scholar 

  • Zuber, J., Pixner, S., Moeller, K., & Nuerk, H.-C. (2009). On the language specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity. Journal of Experimental Child Psychology, 102, 60–77.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Education Sciences, Department of Psychology, University of Duisburg-Essen, Essen, Germany

    Moritz Herzog & Annemarie Fritz

  2. Faculty of Education Centre, Childhood Education Department, University of Johannesburg, Johannesburg, South Africa

    Moritz Herzog

  3. Human Sciences Faculty, University of Potsdam, Potsdam, Germany

    Antje Ehlert

  4. Faculty of Education, Centre for Education Practice Research, University of Johannesburg, Johannesburg, South Africa

    Antje Ehlert & Annemarie Fritz

Authors
  1. Moritz Herzog
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antje Ehlert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Annemarie Fritz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moritz Herzog .

Editor information

Editors and Affiliations

  1. Faculty of Education Sciences, Department of Psychology, University of Duisburg-Essen, Essen, Germany

    Annemarie Fritz

  2. Faculty of Education, Centre for Education Practice Research, University of Johannesburg, Johannesburg, South Africa

    Annemarie Fritz

  3. Departamento de Psicologia, Faculdade de Filosofia e Ciências Humanas, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Vitor Geraldi Haase

  4. Niilo Mäki Institute, Jyväskylä, Finland

    Pekka Räsänen

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Herzog, M., Ehlert, A., Fritz, A. (2019). Development of a Sustainable Place Value Understanding. In: Fritz, A., Haase, V.G., Räsänen, P. (eds) International Handbook of Mathematical Learning Difficulties. Springer, Cham. https://doi.org/10.1007/978-3-319-97148-3_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-97148-3_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-97147-6

  • Online ISBN: 978-3-319-97148-3

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation