## Detecting Mathematics Difficulties

As with any learning difficulty, the earlier it is detected, the better. Studies have shown that using intense tutoring with first graders who display problems with calculations significantly improved their end-of-year achievement in mathematics (Fuchs et al., 2005). The key, of course, is early detection so that interventions can begin as soon as practicable.

## Determining the Nature of the Problem

The first task facing educators who deal with students with mathematics difficulties is to determine the nature of the problem. Obviously, environmental causes require different interventions than developmental causes. Low performance on a mathematics test may indicate that a problem exists, but tests do not provide information on the exact source of the poor performance. Standardized tests, such as the Brigance Comprehensive Inventory of Basic Skills—Revised, are available and provide more precise information on whether the problems stem from deficits in counting, number facts, or procedures.

Educators should examine the degree to which students with mathematics difficulties possess the prerequisite skills for learning mathematical operations. What skills are weak, and what can we do about that? They also should look at the mathematics curriculum to determine how much mathematics is being taught and the types of instructional strategies that teachers are using. Are we trying to cover too much? Are we using enough visual and manipulative aids? Are we developing student strengths and not just focusing on their weaknesses?

## Prerequisite Skills

Examining the nature of mathematics curriculum and instruction may reveal clues about how the school system approaches teaching these topics. A good frame of reference is the recognition that students need to have mastered a certain number of skills before they can understand and apply the principles of more complex mathematical operations. Mathematics educators have suggested that the following seven skills are prerequisites to successfully learning mathematics (Sharma, 2006). They are the ability to

- Follow sequential directions
- Recognize patterns
- Estimate by forming a reasonable guess about quantity, size, magnitude, and amount
- Visualize pictures in one’s mind and manipulate them
- Have a good sense of spatial orientation and space organization, including telling left from right, compass directions, horizontal and vertical directions
- Do deductive reasoning, that is, reason from a general principle to a particular instance, or from a stated premise to a logical conclusion
- Do inductive reasoning, that is, come to a natural understanding that is not the result of conscious attention or reasoning, easily detecting the patterns in different situations and the interrelationships between procedures and concepts

Students who are unable to follow sequential directions, for example, will have great difficulty understanding the concept of long division, which requires retention of several different processes performed in a particular sequence. First one estimates, then multiplies, then compares, then subtracts, then brings down a number; and the cycle repeats. Those with directional difficulties will be unsure which number goes inside the division sign or on top of the fraction. Moving through the division problem also presents other directional difficulties: One reads to the right, then records a number up, then multiplies the numbers diagonally, then records the product down below while watching for place value, then brings a number down, and so on.

### Resources

Fuchs, L. S., & Fuchs, D. (2002). Mathematical problem-solving profiles of students with mathematical disabilities with and without comorbid reading disabilities. Journal of Learning Disabilities, 35, 563–573.

The opening paragraph grabbed my attention. My field has been early literacy and I have seen first hand the extraordinary advances made by grade one students with reading difficulties when they participate in the intensive, one to one remediation provided by Marie Clay’s Reading Recovery. This program identifies the specific underlying problems that are causing reading difficulty and then addresses them. This is exactly what you are advocating for mathematics teaching.

The problem is that these are expensive solutions. Reading Recovery is an expensive program because it requires a fully trained RR teacher to work with only one child at a time, and a classroom teacher who knows how to build on what has been done in RR. Short term planners fail to see the long term benefits of this kind of early intervention. If problems are addressed and resolved in grade one, we see the benefits of the investment in fewer remediation classes in middle school, fewer disengaged students and drop outs who have been unable to take advantage of schooling because of reading difficulties.

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