Mathematics is a complex domain that includes geometry, algebra, arithmetic, statistics, probability, calculus, and arithmetic problem-solving domains, among others. These domains require mobilizing several basic abilities related to memory, decoding, visuospatial, symbols, sense of quantity, and logic. Students who experience mathematical learning difficulties have difficulties with any of the mentioned abilities and coordination. Learning difficulties in math are difficult to diagnose due to the absence of uniform standards to determine the presence or absence of learning difficulties (Karagiannakis et al., 2014). Additionally, psychologists disagree on the definition of these difficulties/disabilities, their prevalence, and operational criteria. This paper describes the case study of five students diagnosed with learning disabilities. It includes a description of the diagnostic testing used, analysis of instructional strategies and math error patterns, and proposed interventions.
Classification of Math Learning Disabilities (MLD) and Its Causes
Two systems are used in processing quantities, including the object tracking system (OTS) and the approximate number system (ANS). The OTS is precise and creates an object file that contains concrete information for all objects observed simultaneously. On the other hand, ANS applies to huge quantities. Deficits in ANS and OTS are classified into five categories, including defective ANS and OTS, defective numerosity-coding, multiple deficits, and access deficit (Karagiannakis et al., 2014). These deficits lead to the formation of a student’s self-identity. Apart from deficits in ANS, MLD also results from deficits in exact quantities, which involves small collections of objects and symbols. The core abilities in these two systems determine a child’s ability to apprehend, quantify, add, or subtract a number of sets without counting. Lastly, core abilities also determine the ability to estimate the magnitude of sets of objects and perform simple numerical operations. These systems will be used to classify students’ MLD in this case study.
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Apart from the five deficits mentioned above, three additional deficits that could be used to associate mathematics disorders with neuropsychological deficits are procedural, semantic memory, and spatial. The procedural deficit is associated with the left hemisphere of the brain and results in a delay in understanding simple arithmetic strategies. The delay may result from deficits in verbal working memory or conceptual knowledge. The semantic memory deficit is also associated with the left hemisphere of the brain and presents as deficits in the retrieval of facts due to a deficit in long-term memory. Lastly, the spatial deficit is associated with the right hemisphere and presents as deficits in the spatial representation of numbers (Karagiannakis et al., 2014). Despite the development of these subtypes to classify the profiles of children with math learning disorders, one child cannot belong to one subtype but exhibits characteristics from different ones.
Definition and Characteristics of MLD
MLD is the incongruity between the mathematics test performance and anticipated performance determined by age, intelligence quotient (IQ), and duration of education. Therefore, Geary (2011) determined that the mathematics test performance of children in the third grade is lower than that of low IQ students at the same grade level. Most research studies categorize students who score below the tenth percentile on standardized mathematics achievement tests as MLD. Individual differences in MLD result from genetic and environmental factors. MLD has various comorbidities, such as attention deficit hyperactivity disorder (ADHD). Children with MLD also exhibit social deficits that arise from severe emotional or behavioral problems.
Case Study Participants
The target population for the case study participants was male students between grades 6 and 8. The study participants were selected through convenience sampling. The convenient sample was derived from consulting grade teachers about some of the students that exhibited the lowest mathematics achievement in each respective grade. The convenience sampling method was used since it was the simplest and most efficient method. Each grade teacher between grades 6 and 8 provided a list of students with the lowest mathematics achievement in the respective grades.
A simple test on students’ abilities to accurately identify number representations was used to distinguish between students with low mathematics achievement and those with MLD. Students in the same grade were asked to determine whether combinations between Arabic sets, number sets, and numerals were equal. The exercise assessed children’s representational systems for small and exact quantities. Typically achieving (TA) children were distinguished from MLD students based on their speed in processing representations. These students were able to process representations of three as fast as representations of two. Another follow-up exercise involved asking children to mentally combine sets of objects and Arabic numerals with matching a target number. Five students were selected from this study based on the confirmation of slow number processing. Out of the five students, two were in sixth grade, two in the seventh grade, and one in the eighth grade. The two students in the sixth grade were Scot and Jayden, both aged 11. The seventh-grade student was Peter, a thirteen-year-old boy. Lastly, Jude and Daniel were selected from the eighth grade and were 13 and 14 years old.
Diagnostic Testing Used
Diagnosis Criteria Used
One of the chief mathematics learning disorders considered in the case study is dyscalculia. Two chief criteria were used to determine the presence of dyscalculia. Any difficulties in processing numbers and quantities, especially for lower grade levels, was a criterion for diagnosis with dyscalculia. When students exhibited difficulties in accurately pairing number with the representing quantities, inadequate understanding of the relationship between numbers and quantities, and difficulties in counting, rapid assessment and naming of small quantities, transcoding, and understanding the place value system, they were diagnosed as having dyscalculia. The second criterion was difficulties in basic arithmetic operations and further mathematical tasks. These difficulties were indicated by a failure to understand computational rules, deficits in retrieving math facts, and a lack of transition from computation, to counting, to non-counting strategies. The difficulties worsen with increasing mathematical complexity.
Clinical Examination and Psychometric Testing
Before performing psychometric testing, the five students were first subjected to a clinical examination. The clinical examination included a physical exam and a standard intelligence test. The presence of MLD in the students was not determined by brain damage, low intelligence, brain disease, or an undetected sight or hearing impairment (Haberstroh & Schulte-Körne, 2019). The standardized intelligence test used for this case study was the Wechsler Intelligence Scale for Children (WISC). The test was administered using a paper-and-pencil for one hour. The five students all had test scores higher than the prescribed average of 100. Scot, Jayden, Peter, Jude, and Daniel scored 99, 94, 92, 103, and 97, respectively. Therefore, low intelligence was eliminated as a cause of MLD. Moreover, a physical exam eliminated the presence of brain disease or damage and hearing and sight impairments from all five students.
Apart from the physical examination, three tests were performed to assess the students' computational skills, mental computation, math fluency, and quantitative reasoning. The tests include the Wechsler Individual Achievement Test IV (WIAT-III) Numerical Operations, the Woodcock-Johnson (WJ) IV Math Fluency Test, the Wechsler Intelligence Test for Children (WISC-V) Arithmetic Subtest, and the WIAT-III Math Problem Solving Subtest (Haberstroh & Schulte-Körne, 2019). WIAT-III determines a child’s ability to perform mathematical operations efficiently and accurately. During this test, the children were given several basic math problems to solve on paper. For the students in the sixth and seventh grade, math problems focused on ratio and proportional relationships. Expressions and equations were tested for students in the eighth grade.
The WJ IV Math Fluency Test was performed to test the students’ ability to recall math facts quickly and accurately. The children were given a written math test of computation problems. They were supposed to deal with as many problems as they could in five minutes. The WISC-V Arithmetic subtest was used to test student’s mental math abilities. The test was administered orally. Lastly, the WIAT-III Math Problem Solving subset was administered to assess children’s problem-solving skills. The test utilized verbal and visual prompts (Haberstroh & Schulte-Körne, 2019). All five students were diagnosed with dyscalculia based on the test results.
Individualized Education Program (IEP)
IEPs are written to ensure that students receive an education based on their individual needs. Teachers, guidance counselors, and parents were involved in the preparation of the IEP. The student’s IEPs were developed based on the Taba-Tyler model/ rational planning model (Voinea & Purcaru, 2015). The first step in creating the IEP was diagnosing student’s needs. The diagnosis was determined based on the diagnostic test results. After the diagnosis, the IEP team developed objectives, which led to the selection and organization of content, learning experiences, and evaluation.
Scot is one of the most challenged students in the sample group. Scot immigrated to the US from Mexico while he was in pre-elementary. At first, his low achievement was associated with the lack of a solid foundation in math skills. However, the diagnosis tests proved that he has dyscalculia. He also exhibits behavior disorders, such as self-harm, when he becomes nervous or shy. Scot lives with his two parents and three siblings and is the second-last born of the family. He scored a 55 on the WJ-V math fluency test. Additionally, he scored 58, 65, and 50 in the WIAT-III Numerical Operations, WISC-V Arithmetic Subtest, and the WIAT-III Math Problem Solving Subtest. Scot exhibited normal mental development for an eleven-year-old boy. Based on these results, Scot is eligible for the IEP, which involves working towards alternative achievement standards using alternative assessments. He showed great uncertainty in mathematics. The goal of the IEP was to develop Scot’s ability to do mathematical calculus, overcome shyness, and develop self-confidence. The Mosaic method was used, which required Scot to teach his peers simple calculation exercises and receive instruction from his peers as well (Voinea & Purcaru, 2015). The selected learning group was selected in a manner to ensure that Scot would experience success, thus feel safe and communicate.
Jayden, an eleven-year sixth-grade student, is an Asian immigrant. He lives with his mother and has a normal mental development for his age. The diagnostic tests indicated that he has dyscalculia. One of the chief challenges that might be associated with his dyscalculia is limited English proficiency. He requires transition services from the normal education program to the IEP. The measurable annual goal for Jayden’s IEP is that he will be proficient in mental calculations and numerical operations at the end of the year. The tour of the gallery method was used (Voinea & Purcaru, 2015). The teacher selected a suitable group that would ensure Jayden experiences success in solving mathematical operations. Each individual in the group solves a problem on a hand-out, which is then displayed. Jayden is then expected to go around explaining how each problem in the hand-out was solved. Daniel and Jayden shared the same issues. Therefore, the tour of the gallery method was also used at Daniel’s grade level.
Peter and Jude were both from the US and exhibited normal mental development for their ages. The two had difficulties in solving at-grade level mathematical problems. They were isolated from their peers. The teacher applied the I know, I want to know, and I have discovered a method to teach math problems.
Instructional Strategies
Cognitive Strategy Instruction
For students with MLD, cognitive strategy instruction can be used to enhance learning and improve performance. It helps students keep up to date on information and promote comprehension of the structure of math problems, especially in word problems. Students are able to focus on the linguistic and semantic information of the word problem. Another strategy that can be used for the students with MLD is the metacognitive strategy, which involves self-regulation, assisting students with planning, and modifying the problem-solving approaches. Researchers have shown that a combination of cognitive and metacognitive strategies is effective in increasing students’ understanding and ability to solve math problems (Krawec et al., 2013). The metacognitive instruction strategy (MSI) is part of the Early Numeracy Intervention program.
The MSI strategy incorporates verbal and visual strategies, especially in solving word problems. It involves a consideration of the critical units and numbers in the question, the question being asked, suitable operations, the computational strategy, and distractable information. Teachers can teach the strategy over three days using multiple examples. On the first day, the first two steps are introduced, and on the second day, the last three steps are taught. An explicit instruction strategy should be used to teach MSI (Krawec et al., 2013). The strategy will ensure proficiency in problem-solving. It will also enhance students’ analysis, representation, execution, and evaluation skills.
Additional Strategies
Apart from cognitive strategy instruction and MSI, teachers can also ensure they maintain consistency and communication in school and at home. Using the same instructional approach ensures that students with MLDs feel safe and confident. Another strategy is the use of concrete objects to teach mathematics concepts, such as the number of seats. Using concrete objects facilitates understanding among students, though it may take longer for others. Another effective strategy is the use of specialized materials, which can help students organize their calculations. For instance, graph papers or lined papers can be used to ensure neat work and emphasis of the key concepts using highlighters. Teachers also need to develop and communicate explicit expectations to their students with MLD. It will reduce confusion and ensure that students do not feel overwhelmed. Teachers can also encourage students to apply mathematical concepts to familiar situations. For example, measurement concepts can be taught by first asking them to estimate their measurements and those of their family members. Teachers should also help students apply the concepts to new situations. Instructors can also use flashcards to list students’ most common errors, which would encourage them to refer to the cards when completing assignments (Heyd-Metzuyanim, 2013). They can also play math games to enhance proficiency.
Math Error Patterns
Dyscalculia can present in the form of operational, spatial, or verbal developmental dyscalculia. Operational dyscalculia is associated with difficulties in planning and executing complex mathematical operations. Spatial dyscalculia involves difficulties in mental calculations and routines, while verbal developmental dyscalculia involves the use of immature strategies and making numerous mistakes in executing complex procedures. Another math error pattern is verbal dyscalculia, which involves wrong associations in retrieval, difficulties with language comprehension, orally presented assignments, passive vocabulary, and conceptual knowledge assignments. Spatial dyscalculia involves the misalignment and misplacements of digits, while graphical dyscalculia involves disturbances in number knowledge (Voinea & Purcaru, 2015). The five students mostly exhibited operational, spatial, and verbal dyscalculia.
Core Content
For sixth grade students with MLDs, mathematical core content involves connecting ratio and rate, division of fractions and manipulation of negative numbers, and understanding statistical thinking. Students are also expected to apply the concepts of rates and ratios to solve mathematical problems. Lastly, sixth-grade students are taught to write, interpret, and use expressions and equations. At the seventh-grade level, mathematical core content includes developing an understanding of proportional relationships and applying them. Students are also expected to understand rational number operations and develop and manipulate expressions and linear equations. Additionally, seventh-grade core content involves understanding two- and three-dimensional shapes and solving area, surface area, and volume problems. They should also learn to draw inferences on populations based on samples. Lastly, eighth-grade core content includes formulating expressions and equations, which also includes modeling and solving linear equations and systems of linear equations. Eighth-grade students should understand how to use functions to describe quantitative relationships. They should also use distance, angle, similarity, and congruence to analyze two- and three-dimensional space and figures.
Proposed Interventions
According to Haberstroh & Schulte-Körne (2019), the most suitable interventions are symptom-specific. A student is assigned mathematical tasks to help them practice in areas they exhibit weakness. Symptom-specific interventions improve mathematical performance in all areas compared to non-symptom specific interventions or no interventions at all. Symptom-specific interventions improve performance in word problems, numerical and quantitative processing, and basic arithmetic operations (Haberstroh & Schulte-Körne, 2019). Therefore, the interventions are more effective than other interventions that focus on enhancing other skills, such as working memory.
For children that struggle with the special representation of numbers, number line training can be used as an effective intervention to enhance number line placement acuity. The child is tasked to write a figure on an empty number line within beginning and endpoints. Children with dyscalculia often make nonlinear placements that adhere to a logarithmic model of placements. The acuity of number line placements attests to the child’s mapping ability between symbolic numbers and non-symbolic quantities. A numerical linear board game can be used to train children on the linear ordering of numbers. The training enhances visual imagery of number information, which precedes comprehension of complex algorithm skills (Michels et al., 2018). Therefore, number line training improves math abilities for children with dyscalculia.
Technology-based interventions, such as the use of basic computer-based play pedagogy intervention, can be used in children displaying dyscalculia characteristics. The games are designed with a dyscalculia-remedy-oriented approach, including number orientation manipulation and repetition. Engaging in the computer game for one hour daily for five days a week reduces number disorientation and arithmetic operation confusion (Mohd Syah et al., 2016). Therefore, schools should embrace technological interventions in designing MLD interventions.
Conclusion
Mathematics is a complex domain that requires mobilizing several basic abilities related to memory, decoding, visuospatial, symbols, sense of quantity, and logic. Students who experience mathematical learning difficulties have difficulties with any of the mentioned abilities and coordination. Mathematics deficits include deficits in ANS and MLD. Other deficits, such as procedural, semantic memory, and spatial, are associated with the left hemisphere and the right hemisphere, respectively. Five students were selected from this study based on the confirmation of slow number processing. The diagnosis was performed using the Wechsler Individual Achievement Test IV (WIAT-III) Numerical Operations, the Woodcock-Johnson (WJ) IV Math Fluency Test, the Wechsler Intelligence Test for Children (WISC-V) Arithmetic Subtest, and the WIAT-III Math Problem Solving Subtest. All five students were diagnosed with dyscalculia based on the test results. The student’s IEPs were developed based on the Taba-Tyler model/ rational planning model and the Mosaic method. Suitable instruction strategies for students with MLD include cognitive strategy instruction, MSI, and symptom-specific interventions. Technology-based interventions can also be used.
References
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