A Psychometric Analysis of the Algebra and Precalculus Concept Readiness Assessment for Community Colleges

Abstract

Understanding students’ readiness for precalculus and calculus at the community college level is critical not only because of the key role community colleges play in higher education but also because calculus remains a gateway course for students in advancing to higher level mathematics. The Algebra and Precalculus Concept Readiness Assessment for Community Colleges (APCR-CC) was designed to investigate community college students’ quantitative reasoning abilities and conceptual understanding in algebra. The present study investigates the psychometric properties of the APCR-CC instrument using item response theory based on a sample of intermediate and college algebra students from six community colleges collected in a pretest (N = 1,131) and posttest (N = 772) setting. We examine unidimensionality, item fit, local item independence, measurement invariance, and sensitivity to instruction. Our findings suggest that the APCR-CC instrument is sufficiently characterized by one underlying construct, local dependence does not seem to be an issue, and 80% of the items in the APCR-CC instrument are sensitive to instruction.

Keywords

algebra and precalculus concept readiness college assessment item response theory psychometric analysis

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References

American Association of Community Colleges. (2017). Fast facts 2017. Washington, DC. Retrieved from https://www.aacc.nche.edu/research-trends/fast-facts/

American Mathematical Association of Two-Year Colleges. (2018). IMPACT: Improving mathematical prowess and college teaching. Memphis, TN: Author.

Blair

Kirkman

E. E.

Maxwell

J. W.

(2018). Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2015 CBMS survey. Providence, RI: American Mathematical Society.

Breidenbach

Dubinsky

Hawks

Nichols

(1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247-285.

Cai

Thissen

du Toit

S. H. C.

(2011). IRTPRO for windows [Computer software]. Lincolnwood, IL: Scientific Software International.

Carlson

Jacobs

Coe

Larsen

Hsu

(2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33, 352-378.

Carlson

Oehrtman

Engelke

(2010). The Precalculus Concept Assessment: A tool for assessing students’ reasoning abilities and understandings. Cognition and Instruction, 28, 113-145.

Champion

Mesa

(2017). Pathways to calculus in U.S. high schools. PRIMUS, 28, 508-527. doi:10.1080/10511970.2017.1315473

Chen

F. F.

(2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 34, 155-175.

10.

de Ayala

R. J

. (2009). The theory and practice of item response theory. New York, NY: Guilford Press.

11.

de Ayala

R. J.

Hertzog

M. A

. (1991). The assessment of dimensionality for use in item response theory. Multivariate Behavioral Research, 26, 765-792. doi:10.1207/s15327906mbr2604_9

12.

Edwards

M. C.

Houts

C. R.

Wirth

R. J.

(2018). Measurement invariance, the lack thereof, and modeling change. Quality of Life Research, 27, 1735-1743.

13.

Haladyna

T. M.

Rodriguez

M. C.

(2013). Developing and validating test items. New York, NY: Routledge.

14.

Harel

Behr

Post

Lesh

(1992). The blocks task: Comparative analyses of the task with other proportion tasks and qualitative reasoning skills of seventh-grade children in solving the task. Cognition and Instruction, 9, 45-96.

15.

Hestenes

Wells

Swackhamer

(1992). Force Concept Inventory. The Physics Teacher, 30, 141-158.

16.

Hoffer

(1988). Ratios and proportional thinking. In Post

T. R.

(Ed.), Teaching mathematics in grades K-8: Research based methods (pp. 285-313). Boston, MA: Allyn & Bacon.

17.

Jang

E. E.

Roussos

(2007). An investigation into the dimensionality of TOEFL using conditional covariance-based nonparametric approach. Journal of Educational Measurement, 44, 1-21.

18.

Kim

E. S.

Cao

Wang

Nguyen

D. T.

(2017). Measurement invariance testing with many groups: A comparison of five approaches. Structural Equation Modeling: A Multidisciplinary Journal, 24, 524-544.

19.

Lamon

S. J.

(2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In Lester

F. K.

(Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Charlotte, NC: Information Age Publishing.

20.

Meade

A. W.

Lautenschlager

G. J.

Hecht

J. E.

(2005). Establishing measurement equivalence and invariance in longitudinal data with item response theory. International Journal of Testing, 5, 279-300.

21.

National Student Clearinghouse Research Center. (2017). The role of community colleges in postsecondary success. Retrieved from https://studentclearinghouse.info/onestop/wp-content/uploads/Comm-Colleges-Outcomes-Report.pdf

22.

Paek

Han

K. T.

(2012). IRTPRO 2.1 for windows (Item response theory for patient-reported outcomes). Applied Psychological Measurement, 37, 242-252.

23.

Pendergast

L. L.

von der Embse

Kilgus

S. P.

Eklund

K. R.

(2017). Measurement equivalence: A non-technical primer on categorical multi-group confirmatory factor analysis in school psychology. Journal of School Psychology, 60, 65-82.

24.

Putnick

D. L.

Bornstein

M. H.

(2016). Measurement invariance conventions and reporting: The state of the art and future directions for psychological research. Developmental Review, 41, 71-90. doi:10.1016/j.dr.2016.06.004

25.

Reckase

M. D.

(1997). The past and future of multidimensional item response theory. Applied Psychological Measurement, 21, 25-36. doi:10.1177/0146621697211002

26.

Robitzsch

(2018). sirt: Supplementary item response theory models. (R package version 2.6-9). Retrieved from https://CRAN.R-project.org/package=sirt

27.

Saldanha

Thompson

P. W.

(1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In Berenson

S. B.

Coulombe

W. N.

(Eds.), Proceedings of the annual meeting of the psychology of mathematics education-North America. Raleigh: North Carolina State University.

28.

Sass

D. A.

Schmitt

T. A.

(2013). Testing measurement and structural invariance: Implications for practice. In Teo

(Ed.), Handbook of quantitative methods for educational research (pp. 315-345). Rotterdam, The Netherlands: Sense Publishers.

29.

Sitomer

Strom

Mesa

Duranczyk

I. M.

Nabb

Smith

Yannotta

(2012). Moving from anecdote to evidence: A proposed research agenda in community college mathematics education. Mathamatyc Educator, 4, 35-40.

30.

Smith

Thompson

P. W.

(2007). Quantitative reasoning and the development of algebraic reasoning. In Kaput

J. J.

Carraher

D. W.

Blanton

M. L.

(Eds.), Algebra in the early grades (pp. 95-132). New York, NY: Erlbaum.

31.

Snyder

T. D.

de Brey

Dillow

S. A.

(2018). Digest of education statistics 2016 (NCES 2017-094). Washington, DC: National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education.

32.

Thompson

P. W.

(1994a). Students, functions, and the undergraduate mathematics curriculum. In Dubinsky

Schoenfeld

A. H.

Kaput

J. J.

(Eds.), Research in collegiate mathematics education, 1 (Vol. 4, pp. 21-44). Providence, RI: American Mathematical Society.

33.

Thompson

P. W.

(1994b). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.

34.

Thompson

P. W.

Saldanha

L. A.

(2003). Fractions and multiplicative reasoning. In Kilpatrick

Martin

W. G.

Schifter

(Eds.), A Research companion to principles and standards for school mathematics (pp. 95-113). Reston, VA: National Council of Teachers of Mathematics.

35.

Wladis

Offenholley

Licwinko

Dawes

Lee

(2018). Development of the elementary algebra concept inventory for the college context. Paper presented at the 21st Annual Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

36.

Zhang

(2007). Conditional covariance theory and DETECT for polytomous items. Psychometrika, 72, 69-91.