## 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.

This sounds like a lot of my students in my 5th grade class. They were like this last year as well as all the previous years. In fact, a good majority of the grade level is like this, and has been since kindergarten. I read that a person suffering from this type of learning challenge will so for the remainder of their lives. So do the studies say how to help an individual cope or work with these challenges? Is there a method I can use to help my students?

I am wondering the same thing as Wendy. How do we help children cope with this challenge? Should we give them math tables and make sure they understand how to find the answer on them? This way, they have a guide to the correct answer. Show them how to follow a guide to get to the correct answer. Use a calculator.

Still, we shouldnt see this problem occurring in whole classes, year after year, should we? That would indicate that something was lacking in the teaching part,

I just stumbled across this blog and have found it very interesting. Like both of you, many of my students have difficulties like those described in this article. Have either of you come across answers to your questions? How do we use this information to help our students?

One thing I have started doing with 2 of the kids I work with is addressing the language of mathematics. Textbooks use many terms to describe the same operation so I try to get a better understanding of what those terms actually mean to the kids. One of my students is fairly good at addition with a number line, but COULD NOT understand subtraction or “counting back” as the book stated. (Her teacher was at her wits end!) The student and i went through a whole conversation about opposites using as many visuals and real life examples….big&small, open&closed….forward&no answer(?)….(movie reference).fast forward& rewind (she was starting to get it), (car reference) reverse&speed ahead. When we were finished I made word wall w these terms, in opposite colors of course, that we review frequently. It has totally made a difference. She still struggles with her facts sometimes (number line works better than a hundreds chart- for her) and has not memorized a single fact beyond y+1, but she demonstrates a better understanding of addition/subtraction.