Advertisement

Assessing Mathematical Competence and Performance: Quality Characteristics, Approaches, and Research Trends

  • Jan LonnemannEmail author
  • Marcus Hasselhorn
Chapter

Abstract

To assess children’s individual stages of mathematical development, as well as their developmental trajectories, different diagnostic approaches have been developed. Based on the test results of such approaches, children with mathematics learning difficulties are identified and programs to enhance mathematical learning can be developed and evaluated. In this chapter, quality characteristics of assessment are described (reliability, objectivity, validity, and provision of norms). Different categories are presented to classify approaches to assessing mathematical competence and performance (norm-referenced versus not-norm-referenced tests, individual versus group testing, paper-and-pencil tests versus interviews versus computer-based tests, chronological versus educational age–oriented tests, speed versus power tests, and principles of task selection). Drawing on selected approaches, different principles of task selection are discussed (curriculum-based, based on neuropsychology theories, or based on developmental psychology theories) together with their consequences for the interpretation of respective test results. Finally, some promising research trends are outlined.

Keywords

Mathematical competence Mathematical performance Learning difficulties Psychological assessment Diagnostics 

References

  1. Aunio, P. (2006). Number sense in young children—international group differences and an intervention programme for children with low and average performance (research report 269). Helsinki, Finland: University of Helsinki.Google Scholar
  2. ASER Centre. (2017). In ASER Centre (Ed.), Annual status of education report (rural) 2016. New Delhi, India.Google Scholar
  3. Butterworth, B. (2003). Dyscalculia screener. London: NFER-Nelson.Google Scholar
  4. Clements, D. H., Sarama, J. H., & Liu, X. H. (2008). Development of a measure of early mathematics achievement using the Rasch model: the research-based early maths assessment. Educational Psychology, 28, 457–482.CrossRefGoogle Scholar
  5. Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120.Google Scholar
  6. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42.CrossRefGoogle Scholar
  7. Dellatolas, G., von Aster, M., Willardino-Braga, L., Meier, M., & Deloche, G. (2000). Number processing and mental calculation in school children aged 7 to 10 years: transcultural comparison. European Child & Adolescent Psychiatry, 9, 102–110.CrossRefGoogle Scholar
  8. Deno, S. L. (1985). Curriculum-based measurement: the emerging alternative. Exceptional Children, 52, 219–232.CrossRefGoogle Scholar
  9. Foegen, A., Jiban, C., & Deno, S. (2007). Progress monitoring measures in mathematics: a review of the literature. The Journal of Special Education, 41, 121–139.CrossRefGoogle Scholar
  10. Fritz, A., Ehlert, A., Ricken, G., & Balzer, L. (2017). Mathematik und Rechenkonzepte im ersten Schuljahr – Diagnose (MARKO-D1+) [Assessment of math concepts in first graders—diagnosis (MARKO-D1+)]. Göttingen, Germany: Hogrefe.Google Scholar
  11. Fritz, A., Ehlert, A., & Balzer, L. (2013). Development of mathematical concepts as basis for an elaborated mathematical understanding. South African Journal of Childhood Education, 3, 38–67.Google Scholar
  12. Fritz, A., & Ricken, G. (2008). Rechenschwäche [Mathematics learning difficulties]. München, Germany: Reinhardt.Google Scholar
  13. Fuchs, L. S., Hamlett, C. L., & Fuchs, D. (1998). Monitoring basic skills progress: math computation. Austin, TX: Pro-Ed.Google Scholar
  14. Fuson, C. K. (1988). Children’s counting and concepts of number. New York: Springer.CrossRefGoogle Scholar
  15. Geary, D. C. (2000). From infancy to adulthood: the development of numerical abilities. European Child & Adolescent Psychiatry, 9, 11–16.CrossRefGoogle Scholar
  16. Gerlach, M., Fritz, A., & Leutner, D. (2013). MARKO-T. Mathematik- und Rechenkonzepte im Vorschul- und Grundschulalter Training [Mathematics and arithmetic training for developing concepts in preschool and early primary-school age]. Göttingen, Germany: Hogrefe.Google Scholar
  17. Gersten, R., Clarke, B., Jordan, N. C., Newman-Gonchar, R., Haymond, K., & Wilkins, C. (2012). Universal screening in mathematics for the primary grades: beginnings of a research base. Exceptional Children, 78, 423–445.CrossRefGoogle Scholar
  18. Ginsburg, H. P., & Baroody, A. J. (2003). The test of early mathematics ability (3rd ed.). Austin, TX: Pro-Ed.Google Scholar
  19. Goo, M., Watt, S., Park, Y., & Hosp, J. (2012). A guide to choosing web-based curriculum-based measurements for the classroom. Teaching Exceptional Children, 45, 34–40.CrossRefGoogle Scholar
  20. Grégoire, J., Noël, M., & Nieuwenhoven, v. (2004). Tedi-Math (Flemish adaptation). Antwerpen, Belgium: TEMA: Brussels/Harcourt.Google Scholar
  21. Henning, E., Ehlert, A., Ragpot, L., Herholdt, R., Balzer, L., & Fritz, A. (in press). MARKO-D-South-Africa: assessment of math concepts in first graders. Johannesburg, South Africa: University of Johannesburg.Google Scholar
  22. Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (2016). Handbook of response to intervention: the science and practice of multi-tiered systems of support. Springer Science & Business Media.Google Scholar
  23. Jordan, N. C., & Glutting, J. (2012). Number sense screener (K-1): research edition. Baltimore: Paul H. Brookes Publishing.Google Scholar
  24. Kaufmann, L., Nuerk, H.-C., Graf, M., Krinzinger, H., Delazer, M., & Willmes, K. (2008). Test zur Erfassung numerisch-rechnerischer Fertigkeiten vom Kindergarten bis zur 3. Klasse (Tedi-Math) [Test for the assessment of numerical and calculation skills from kindergarten age to third grade (Tedi-Math)]. Bern, Switzerland: Huber.Google Scholar
  25. Koumoula, A., Tsironi, V., Stamouli, V., Bardani, I., Siapati, S., Graham, A., et al. (2004). An epidemiological study of number processing and mental calculation in Greek schoolchildren. Journal of Learning Disabilities, 37, 377–388.CrossRefGoogle Scholar
  26. Krajewski, K. (2008). Prävention der Rechenschwäche [The early prevention of math problems]. In W. Schneider & M. Hasselhorn (Eds.), Handbuch der Pädagogischen Psychologie (pp. 360–370). Göttingen, Germany: Hogrefe.Google Scholar
  27. Krajewski, K., Küspert, P., Schneider, W., & Visé, M. (2002). Deutscher Mathematiktest für erste Klassen (DEMAT 1+) [German mathematics tests for first grade]. Göttingen, Germany: Hogrefe.Google Scholar
  28. Krajewski, K., & Schneider, W. (2009). Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: findings from a four-year longitudinal study. Learning and Instruction, 19, 513–526.CrossRefGoogle Scholar
  29. Montague, M., Penfield, R. D., Enders, C., & Huang, J. (2010). Curriculum-based measurement of math problem solving: a methodology and rationale for establishing equivalence of scores. Journal of School Psychology, 48, 39–52.CrossRefGoogle Scholar
  30. Renaissance Learning. (2012). STAR-math (enterprise). Wisconsin Rapids: Renaissance Learning. http://www.renaissance.com/products/star-assessments
  31. Resnick, L. B. (1983). A developmental theory of number understanding. In H. Ginsburg (Ed.), The development of mathematical thinking (pp. 109–151). New York: Academic.Google Scholar
  32. Ricken, G., Fritz, A., & Balzer, L. (2013). MARKO-D—Mathematik- und Rechnen- Test zur Erfassung von Konzepten im Vorschulalter [MARKO-D—mathematics and arithmetic test for assessing concepts at preschool age]. Göttingen, Germany: Hogrefe.Google Scholar
  33. Santos, F. H., Da Silva, P. A., Ribeiro, F. S., Dias, A. L. R. P., Frigério, M. C., Dellatolas, G., et al. (2012). Number processing and calculation in Brazilian children aged 7–12 years. The Spanish Journal of Psychology, 15, 513–525.CrossRefGoogle Scholar
  34. Spelke, E., Lee, S. A., & Izard, V. (2010). Beyond core knowledge: natural geometry. Cognitive Science, 34, 863–884.CrossRefGoogle Scholar
  35. Schrank, F. A., McGrew, K. S., & Mather N. (2015). Woodcock-Johnson IV Tests of Early Cognitive and Academic Development. Rolling Meadows, IL: Riverside.Google Scholar
  36. Uwezo. (2014). http://www.uwezo.net/
  37. van Luit, J. E. H., Van de Rijt, B. A. M., & Aunio, P. (2006). Early numeracy test, Finnish edition. Helsinki, Finland: Psykologien Kustannus.Google Scholar
  38. van Luit, J. E. H., Van de Rijt, B. A. M., & Hasemann, K. (2001). OTZ Osnabrücker Test zur Zahlbegriffsentwicklung [German version of the Utrecht test of number sense]. Göttingen, Germany: Hogrefe.Google Scholar
  39. van Luit, J. E. H., van de Rijt, B. A. M., & Pennings, A. H. (1994). Utrechtse Getalbegrip Toets [Utrecht test of number sense]. Doetinchem, The Netherlands: Graviant.Google Scholar
  40. van Nieuwenhoven, C., Grégoire, J., & Noël, M. P. (2001). Tedi-Math. In Test Diagnostique des Compétences de Base en Mathématiques [Tedi-Math: diagnostic test for basic competencies in mathematics]. Paris: Editions Centre de Psychologie Appliquée à Paris.Google Scholar
  41. von Aster, M. G., Bzufka, M. W., & Horn, R. R. (2009). Neuropsychologische Testbatterie für Zahlenverarbeitung und Rechnen bei Kindern—Kindergartenversion—(ZAREKI-K) [Neuropsychological test battery for number processing and calculation in children—kindergarten version—(ZAREKI-K)]. Frankfurt am Main, Germany: Pearson.Google Scholar
  42. von Aster, M., & Dellatolas, G. (2006). ZAREKI-R: batterie pour l’évaluation du traitement des nombres et du calcul chez l’enfant. Adaptation Francaise [ZAREKI-R: neuropsychological test battery for number processing and calculation in children. French adaptation]. Paris: ECPA.Google Scholar
  43. von Aster, M. G., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49, 868–873.CrossRefGoogle Scholar
  44. von Aster, M. G., Weinhold Zulauf, M., & Horn, R. (2006). Neuropsychologische Testbatterie für Zahlenverarbeitung und Rechnen bei Kindern (ZAREKI-R) [Neuropsychological test battery for number processing and calculation in children (ZAREKI-R). Frankfurt am Main, Germany: Pearson.Google Scholar
  45. Wechsler, D. (2009). Wechsler individual achievement test (3rd edition). San Antonio, TX: Pearson.Google Scholar
  46. Wechsler, D. (2010). Wechsler individual achievement test—third edition: Canadian (WIAT-III CDN). Toronto, ON: Pearson.Google Scholar
  47. Wechsler, D. (2016). Wechsler individual achievement test—Australian and New Zealand standardised, third edition (WIAT-III A&NZ). Sydney, Australia: Pearson.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Empirical Childhood ResearchUniversity of PotsdamPotsdamGermany
  2. 2.Department of Education and Human DevelopmentGerman Institute for International Educational Research (DIPF)Frankfurt am MainGermany
  3. 3.Center for Individual Development and Adaptive Education of Children at Risk (IDeA)PotsdamGermany
  4. 4.German Institute for International Educational Research (DIPF)Frankfurt am MainGermany
  5. 5.Department of Educational PsychologyInstitute for Psychology, Goethe-UniversityFrankfurt am MainGermany

Personalised recommendations