## Types of Mathematical Disorders

The complexity of mathematics makes the study of mathematical disorders particularly challenging for researchers. Learning deficits can include difficulties in mastering basic number concepts, counting skills, and processing arithmetic operations as well as procedural, retrieval, and visual-spatial deficits (Geary, 2004). As with any learning disability, each of these deficits can range from mild to severe.

**Number concept difficulties.** An understanding of small numbers and quantity appears to be present at birth. The understanding of larger numbers and place value, however, develops during the preschool and early elementary years. A poor understanding of the concepts involved in a mathematical procedure will delay the adoption of more sophisticated procedures and limit the child’s ability to detect procedural errors. Studies show that most children with mathematical disorders nevertheless have their basic number competencies intact. However, they often are unable to use their number concept skills to solve arithmetic problems (Geary, 2004).

**Counting skill deficits.** Studies of children with mathematical disorders show that they have deficits in counting knowledge and counting accuracy. Some may also have problems keeping numerical information in working memory while counting, resulting in counting errors.

**Difficulties with arithmetic skills.** Children with mathematical disorders have difficulties solving simple and complex arithmetic problems, and they rely heavily on finger counting. Their difficulties stem mainly from deficits in both numerical procedures (solving 6 + 5 or 4 × 4) and working memory. They tend to use developmentally immature procedures, such as counting all rather than counting on.

At the same time, they do not show the shift from procedure-based problem solving to memory-based problem solving that is found in typically achieving children, most likely because of difficulties in storing arithmetic facts or retrieving them from long-term memory. Moreover, deficits in visual-spatial skills can lead to problems with arithmetic because of misalignment of numerals in multi-column addition. Although procedural, memory, and visual-spatial deficits can occur separately, they are often interconnected.

**Procedural disorders.** Students displaying this disorder:

- Use arithmetic procedures (algorithms) that are developmentally immature
- Have problems sequencing multistep procedures, such as 52 × 13 or 317 + 298
- Have difficulty understanding the concepts associated with procedures
- Make frequent mistakes when using procedures

The exact cause of this disorder is unknown, but research studies have yielded some intriguing findings. Children with developmental or acquired dyscalculia can still count arrays of objects, say the correct sequence of number words while counting, and understand basic counting concepts, such as cardinality. However, they have difficulties in solving complex arithmetic problems. Researchers suspect one possible cause may be a dysfunction in the brain’s left hemisphere, which specializes in procedural tasks.

**Memory disorders**. Students displaying this disorder:

- Have difficulty retrieving arithmetic facts
- Have a high error rate when they do retrieve arithmetic facts
- Retrieve incorrect facts that are associated with the correct facts
- Rely on finger counting because it reduces the demands on working memory

This disorder likely involves the manipulation of information in the language system. Here again, a dysfunction of the left hemisphere is suspected, mainly because these individuals frequently have reading disorders as well (D’Amico & Guarnera, 2005). This association further suggests that memory deficits may be inheritable.

Memory disorders can be caused by two separate problems. One involves disruptions in the ability to retrieve basic facts from long-term memory, resulting in many more errors than typically achieving children. Research findings indicate that this form of memory disorder is closely linked to the language-processing system and may indicate developmental or acquired deficits in the left hemisphere.

The second possibility involves disruption in the retrieval process caused by difficulties in inhibiting the retrieval of irrelevant associations. Thus the student seems impulsive. For example, when asked what is 7 + 3, a student might quickly blurt out 8 or 4 because those numbers come next in counting (Passolunghi & Siegel, 2004). Solving arithmetic problems becomes much easier when irrelevant information is prevented from entering working memory. When irrelevant information is retrieved, it lowers working memory’s capacity and competes with correct information for the individual’s attention. This type of retrieval deficit may be caused by deficits in the brain’s executive areas of the prefrontal cortex responsible for inhibiting working-memory operations.

**Visual-spatial deficits**. Students with this disorder:

- Have difficulties in the spatial arrangement of their work, such as aligning the columns in multicolumn addition
- Often misread numerical signs, rotate and transpose numbers, or both
- Misinterpret spatial placement of numerals, resulting in place value errors
- Have difficulty with problems involving space in areas, as required in algebra and geometry

Studies indicate that this disorder is closely associated with deficits in the right parietal area, which specializes in visual-spatial tasks. Individuals with injuries to this area often show a deficit in spatial orientation tasks and in the ability to generate and use a mental number line (Zorzi, Priftis, & Umiltá, 2002). Some studies suggest that the left parietal lobe also may be implicated.

Many students eventually overcome procedural disorders as they mature and learn to rely on sequence diagrams and other tools to remember the steps of mathematical procedures. Those with visual-spatial disorders also improve when they discover the benefits of graph paper and learn to solve certain algebra and geometry problems with logic rather than through spatial analysis alone. However, memory deficits do not seem to improve with maturity. Studies indicate that individuals with this problem will continue to have difficulties retrieving basic arithmetic facts throughout life. This finding may suggest that the memory problem exists not just for mathematical operations, but may signal a more general deficit in retrieving information from memory.

### Resources:

D’Amico, A., & Guarnera, M. (2005). Exploring working memory in children with low arithmetic achievement. Learning and Individual Differences, 15, 189–202.

Geary, D. C. (2004, January-February). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15.

Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access to numerical information in children with disability in mathematics. Journal of Experimental Child Psychology, 88, 348–367.

Zorzi, M., Priftis, K., & Umiltá, C. (2002). Neglect disrupts the mental number line. Nature, 417, 138.

## Students With Both mathematics and Reading Difficulties

Students who have both reading and mathematics difficulties are obviously at a double disadvantage. However, even though the reading and mathematical processing areas of the brain are separate from each other, these two cerebral regions interact whenever the learner must translate word problems into symbolic representations. Here are some strategies that are effective with these students.

**Cue words in word problems**. Help these students decode language into mathematical operations by alerting them to common phrases or cue words found in word problems that identify which operation to use.

**Word problem maps.** Give students with reading problems a story map to highlight certain important aspects of the story such as introduction, plot line, characters, time line, and story climax. Gagnon and Maccini (2001) have developed a similar learning aid, called a word problem map, to help students with mathematics difficulties organize their thoughts as they tackle word problems. The map can be completed by an individual student or by students working in groups of two or three.

**The RIDD strategy**. The RIDD strategy was developed by Jackson (2002) in 1997 for students with learning disabilities. In practice, it has shown to be particularly helpful to students who have difficulties in both reading and mathematics. RIDD stands for Read, Imagine, Decide, and Do. The following is a description of these four steps.

** Step 1:** Read the problem. Read the passage from beginning to end. This helps students focus on the entire task rather than just one line at a time. Good readers often skip words within a text, or they substitute another word and continue reading. In this step, students decide ahead of time what they will call a word that they do not recognize. In mathematics word problems, substitutions can be made for long numbers rather than saying the entire number on the first reading. Teachers should model this substitution when they read the problem aloud to the class.

**Step 2**: Imagine the problem. In this step, the students create a mental picture of what they have read. Using imagery when learning new material activates more brain regions and transforms the learning into meaningful visual, auditory, or kinesthetic images of information. This makes it easier for the new information to be stored in the students’ own knowledge base. Imagery helps students focus on the concept being presented, and provides a way of monitoring their performance.

**Step 3**: Decide what to do. In order to generate a mental picture of the situation, this step encourages students to read the entire mathematics problem without stopping. They then decide what to do and in what order to solve the problem. For example, in a word problem requiring addition and then subtraction, students would read the problem, create a mental picture, and then decide whether to add or subtract first. For young students, teachers can guide them through this step with appropriate questioning so the students can decide what procedures to use. Note how this step combines reading, visualization, and problem solving.

**Step 4:** Do the work. During this step the students actually complete the task. Often, students start reading a mathematics problem, stop part way through it, and begin writing numerical expressions. This process can produce errors because the students do not have all the information. By making this a separate step, students realize that there are things to do between reading the problem and writing it down. Jackson (2002) observed that when students used RIDD to solve mathematics problems, they liked this strategy because they perceived the last step as the only time they did work. Apparently, the students did not realize that what they did in the first three steps was all part of the process for solving problems.

**Computer assistance.** Computer programs are now available for elementary level students that address both reading and mathematics weaknesses. For example, Knowledge Adventure has several software titles that focus on teaching basic mathematics and reading skills while adhering to national and state standards. Each program provides instruction at a student’s own pace and includes automatic progress tracking for each student so teachers can provide additional instruction to those who need it.

**References:**

Gagnon, J., & Maccini, P. (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 34, 8–15.

Jackson, F. B., (2002, May). Crossing content: A strategy for students with learning disabilities. Intervention in School and Clinic, 37, 279–282.