Students’ proficiency in mathematics is a concern as students in the United States continue to be outperformed by peers in other industrialized nations on assessments of mathematics achievement (NMAP, 2008).  Success in mathematics is linked to graduation, higher education, and employment (NMAP, 2008; NCTM 2010), yet results from national and international reports alike suggest students in our educational system may lack skills necessary to compete in arenas requiring proficiency in mathematics.

            It has been suggested that students must reach proficiency in pre-requisite skills, such as operations involving fractions, prior to entering high school (NMAP, 2008).   Consequently, growing consensus among experts indicates students need strong curricula and effective instruction, grounded in empirical research (e.g., NMAP, 2008).  An example of an evidence-based practice that has been effectively used to teach mathematics to students who chronically struggle with mathematics is Concrete-Representational-Abstract (CRA) sequence of instruction.  The presentation will address the priority area of instructional interventions by sharing research and describing an evidence-supported intervention.  This poster session will share results from an experimental study comparing CRA sequences instruction to traditional instruction of fractions.

Educational Significance

            Learning naturally progresses from physical representation, anchored in real-world experiences, to abstract understandings (NMAP, 2008).  For some learners, the transition between physical and abstract understanding of mathematics may be seamless and appear effortless, while for other learners, the leap from physical to abstract understanding of mathematics may be too broad and therefore strained.  Struggling learners may therefore benefit from a sequence of instruction that links physical objects to pictorial representations and then links pictorial representations to abstract concepts.  Essentially, pictorial representations, scaffolds the learner from physical to abstract understanding of mathematics.

The body of empirical research supporting for the use of manipulatives with instruction for students who chronically struggle with mathematics is still growing (Witzel & Allsopp, 2007).  CRA sequence of instruction has been used to teach mathematics concepts such as basic facts and area and perimeter skills to struggling learners (Mercer & Miller, 1993; Cass, Cates, Smith, & Jackson, 2003).  CRA sequence of instruction may contribute to long-term maintenance of skills (Cass et al.)  Although CRA has a strong body of research that supports its use in mathematics (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Witzel, 2005; Witzel, Mercer, & Miller, 2003; Flores, 2009), few studies have been conducted to look at acquisition of fractions for middle school students at risk for failure in mathematics.  The current study looked to extend the growing body of research by analyzing CRA instruction on achievement of operations involving fractions for middle school students.

Theoretical Framework and Key Findings

            Vygotsky asserted that individuals learn in social context and acquisitions of new skills are supported by social learning (1978).  Vygotsky suggests students acquire knowledge best within their zone of proximal development (ZPD).  The ZPD is the distance between what a student can do independently versus the potential of what the student can do under guidance and/or with collaboration of a mentor.  The ZPD is the area where students most benefit from instruction because they are introduced to skills that may be too difficult for them to successfully complete by themselves, but can achieve with the scaffolded guidance of an expert individual.  The ZPD recognizes the sequence of development learning and addresses the need to guide students as they learn more complex and abstract concepts.  Vygotsky’s theories are embedded within multiple elements of CRA instruction.  Students are introduced to more abstract and difficult learning only after they have mastered and internalized easier and more concrete representations of the strategy.

Methods

            Thirty-six middle school students with from a school in a southeastern state participated in this study.  An experimental 2X3 design was implemented to evaluate the effect of CRA instruction on acquisition and retention of operations involving fractions.  Participants were randomly assigned to a treatment or control group.  The treatment group received CRA instruction across 30 lessons on operations involving fractions, while the control group received traditional instruction on operations involving fractions for the same amount of time.  The control and treatment groups received the assessment prior to implementing the intervention, after the treatment group has received the six-week CRA intervention, and at a follow-up meeting six weeks after the completion of the intervention.

Results

            The research compared the effects of CRA sequence of instruction with traditional instruction on acquisition and retention of mathematics knowledge involving fractions.  Both groups yielded significant gains from pre to post assessments, yet there were no significant differences between the performances of the two groups.  The delayed-post assessment evaluated retention of fractions skills.  An interaction effect were detected for performance on the delayed-post assessment, F(1,33) = 10.06, p = .003, h^2 = .234, where students in the CRA treatment group significantly outperformed the control group on the assessment.

            Results from this research corroborate findings from previous research that support CRA sequenced instruction for students who are at risk of failure in mathematics. Results from this study differ from previous research, however as initial student performance did not differ on computations of operations involving fraction, however did show significant differences in retention of learned information.  Considering the retention of prerequisite and foundational skills (e.g., rational numbers and fractions) are essential to build a foundation from which students can succeed in mathematics, students who accurately understand concepts and procedures and retain fractions information for longer periods of time may subsequently build for success on future mathematics performance.

Timeline

             A poster session will allow for maximum interaction with participants.  The presenters will provide a brief overview of the topic, description of the intervention, and summary of the results (apx. 5-10 minutes for each group) followed by time for participants to ask questions.  The presenters are prepared to address the needs and questions of the participants as they browse our poster.  Participants will be encouraged to interact with the manipulatives used in the research.  Visual supports will be present on the poster and as additional references. 

References

Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce. T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Practice, 18(2), 99-111.

Cass, M., Cates, D., Smith, M., & Jackson, C. (2003). Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. Disabilities Research & Practice, 18(2), 112-120.

Flores, M. M. (2009). Teaching subtracting with regrouping to students experiencing difficulty in mathematics. Preventing School Failure, 53(3), 145-152.

Mercer, C. D., & Miller, S. P. (1993).  Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial & Special Education, 13(3), 19-35.

National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington DC: U.S. Department of Education.

National Council of Teachers of Mathematics. (2010).  Retrieved on June 15, 2010.  http://www.nctm.org/about/content.aspx?id=210&LangType=1033 retrieved 2-19-10 at 10:15.

Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings.

Witzel, B. S., Mercer. C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research and Practice, 18, 121-131.

Witzel, B. S., & Allsopp, D. (2007). Dynamic concrete instruction in an inclusive classroom. Mathematics Teaching in Middle School, 13(4), 244-248.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

U.S. Department of Education, National Center for Educational Statistics. (2009). Digest of Educational Statistics, 2008 (NCES 2009-020).  Retrieved from http://nces.ed.gov/fastfacts/display.asp?id=64.