Long, H., Bouck, E., & Domka, A. (2021). Manipulating algebra: Comparing concrete and virtual algebra tiles for students with intellectual and developmental disabilities . Exceptionality , 29 (3), 197–214. https://doi.org/10.1080/09362835.2020.1850454

Journal Article

Long, H., Bouck, E., & Domka, A. (2021). Manipulating algebra: Comparing concrete and virtual algebra tiles for students with intellectual and developmental disabilities. Exceptionality, 29(3), 197–214. https://doi.org/10.1080/09362835.2020.1850454


Intellectual disabilities; K-12; Manipulatives; Math; Middle school; U.S. context





Concrete and virtual manipulatives were investigated. These were described as labeled tiles or electronic notebook application with virtual tiles for calculating and otherwise keeping track of addition and subtraction problems.


Three middle school students with intellectual disabilities in grade 8, aged 14–15. Students were all assigned to a self-contained special education class in the same school in a rural town in a midwestern state (U.S.). All three students previously showed proficiency in solving single digit calculations, but scored below 20% accuracy on the researchers' algebra pre-assessment. Inclusion criteria also indicated that participants had the "evaluated or teacher-stated ability to solve single-digit addition, subtraction, multiplication, and division problems with or without a calculator" (p. 200). Pre-screening tools included Wechsler Intelligence Scale for Children Fifth Edition (WISC-V; Wechsler, 2014) for full-scale IQ scores and the KeyMath3™ assessment (Connolly, 2007) which provided grade equivalencies of grade 2 to grade 3 for participants.

Dependent Variable

Student participants completed a researcher administered algebra pre-assessment with less than 20% accuracy. Three factors were tracked throughout the study: (a) observation of their level of independence in using both concrete and virtual manipulatives, (b) task completion time, and (c) accuracy in solving two-step linear algebra problems. Students also participated in social validity interviews, responding to questions such as "(a) what manipulative type was your favorite and why? (b) did both types manipulative help you solve the problems? (c) did you like using the manipulatives? and (d) would you use either type of manipulative again?" (p. 205).


Students with intellectual disabilities in grade 8 were equally independent and accurate, and used similar times, when using concrete (physical) manipulatives and virtual manipulatives. Concrete and virtual manipulatives were equally effective in supporting participants’ performance on algebra test-like tasks. The three participants demonstrated similar benefits when applying the two types of manipulatives during grade-level math tasks over time. All three participants indicated that they liked using the manipulatives, with a slight preference for the concrete tiles. Implications for practice were offered, limitations of the study were reported, and future research possibilities were suggested.