Math Learning Disabilities: Bridging Informal Knowledge to Formal Math

Introduction

Solving problems in mathematics is often regarded as a significant challenge for students in elementary school. When dealing with mathematical questions, students frequently need to employ a range of different skills, and many face difficulties doing so — particularly in areas such as spatial relations, language, and memory. Proficiency in single-digit mathematics is often believed to rest on an increasing dependence on memorized facts. A different model, however, argues that mastering single-digit arithmetic can also be achieved through the progressive use of efficient calculation strategies.

Many young students who struggle with elementary mathematical problems often arrive at school already possessing a solid foundation of informal mathematical understanding. Such children frequently have difficulty not with the concepts themselves, but with relating what they already know to the formal language, procedures, and symbolic notations of academic mathematics. It is genuinely difficult for a child to comprehend an entirely new world of symbols and notations representing quantities and operations, while simultaneously acquiring the specialized language used to teach arithmetic.

These young children therefore need repeated exposure — hearing the words, and using the symbols and notations in many different ways — before they can begin to correctly understand spatial relations, the language of arithmetic, and formal notation systems. Instead of providing these repeated experiences, however, teachers sometimes compound the difficulty by asking children to match numerals with pictured groups before those students have gained adequate experience correlating different quantity representations. As a result, students often need more time to reach understanding than they would if instruction were organized in a manner that built explicitly on what they already know.

Working Memory and Mathematical Fluency

Working memory is a finite-capacity system that allows the temporary storage and processing of small amounts of information simultaneously. It partly explains why learners who lack automatic recall of math facts struggle with problem-solving: skills that are practiced repeatedly and performed fluently place fewer demands on working memory than skills that a student is still acquiring. According to Alter (2012), students who can automatically retrieve key facts from memory are able to devote more working memory resources to the conceptual, language, and procedural dimensions of a mathematical problem. In other words, students solve arithmetic problems more effectively when they are not simultaneously expending cognitive effort on basic counting. Initiating interventions aimed at building math fluency is therefore one of the most effective approaches to addressing difficulties in learning mathematics.

Components of Effective Fluency Interventions

Alter (2012) identified several components he considered essential for high-quality fluency interventions. These include: corrective teaching procedures, immediate feedback, opportunities for students to respond in class, the use of material calibrated to the students' cognitive capacities, and — most importantly — direct instruction. Using appropriately challenging material refers to the ratio of known to unknown items that a learner encounters, based on prior mastery. Interventions that incorporate both known and unknown items are believed to increase a student's chances of success, make tasks more reinforcing, and reduce reliance on external motivators such as rewards, praise, or reprimands (Alter, 2012). For most intervention approaches, a ratio of 70% to 90% known items is considered optimal.

Providing students with regular opportunities to respond in class is equally critical. These are the moments when learners are prompted to produce correct answers, and students need many such opportunities to effectively consolidate new information. Direct instruction, structured to move from simple to complex and to engage students actively, underpins all of these components.

Conclusion

It is important to note that mastering mathematics is linked to the child's increasing dependence on working memory skills. The interventions noted here are only a handful, but there are many more that are sequential in nature and make mathematics engaging to learn. Tapping into young children's informal knowledge base and helping them connect what they already know to what they are expected to learn is central to the mathematical learning process. When instruction is designed with this connection in mind — through fluency-building activities, appropriately paced language instruction, and student-centered engagement — students with math learning disabilities are far better positioned to succeed.

References

Alter, P. (2012). Helping students with emotional and behavioral disorders solve mathematics word problems. Taylor & Francis Group, LLC. http://dx.doi.org/10.1080/1045988X.2011.565283

Garnett, K. (1998). Math learning disabilities. Division for Learning Disabilities.

Jitendra, A., & Star, J. (2011). Meeting the needs of students with learning disabilities in inclusive mathematics classrooms: The role of schema-based instruction on mathematical problem-solving. The College of Education and Human Ecology, The Ohio State University, 50, 12–19. http://dx.doi.org/10.1080/00405841.2011.534912

Prado, J., Mutreja, R., & Booth, J. (2014). Developmental dissociation in the neural responses to simple multiplication and subtraction problems. 17(4), 537–552.