### What Are the Characteristics of Students with Learning Difficulties in Mathematics?

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Researchers have extensively studied students who experience significant problems in their acquisition of mathematical knowledge across multiple school years, regardless of their motivation, the quality of their former mathematics instruction, and their number knowledge and number sense when entering school. Several consistent findings have emanated from this body of research.

*Slow or Inaccurate Retrieval of Basic Arithmetic Facts*

The one bedrock problem found in the literature about students with mathematics difficulties was their extremely slow retrieval of even the most rudimentary arithmetic facts (Hasselbring, Bransford, and Goin 1988). This finding was manifested repeatedly in a number of studies. In particular, one study showed that this problem seemed to persist even as these students progress through elementary school mathematics regardless of their increased proficiency with computation (Geary 1993).

In the early grades, this computation would include facts as simple as 3 + 6. For older students, it might entail simple multiplication and division facts, such as 4 × 2. Students who cannot retrieve these basic facts easily get lost and often cannot follow the logic of an explanation given by the teacher or a peer when the problems are embedded within more complex mathematical operations, such as simple algebra or long division. The teacher or the textbook assumes virtually automatic retrieval of these facts and bases explanations on this assumption. For example, during a lesson on equivalent fractions, teachers will assume that students “just know” that 2 × 5 is 10 and can use this information to follow an explanation of why five tenths is the same as one-half. If a student must use her fingers to count this out, the whole point of the lesson may well be lost.

There is no consensus on how best to assist students with this problem. Most efforts entail daily work on number families to see relationships between facts. The hope is that repeated practice or overlearning will improve retrieval speed. Unfortunately, no one has studied this issue systematically. Teachers need to recognize that some students will have difficulty with quick retrieval and that teachers will need to use alternative means so that students fully understand the concepts presented.

*Impulsivity*

A major problem found in the literature was impulsivity or a lack of inhibition. An example offered by Geary (2005) and Passolunghi and Siegel (2004) helped illustrate this problem. When asked what 4 + 8 is, a student might impulsively blurt out 5 or 9 because those numbers come next when counting. This example shows that the student has great difficulty with inhibiting irrelevant associations and with focusing on the problem at hand. This characteristic may explain why, as will be seen, instructional approaches that prompt students to think aloud or draw out a problem might be particularly helpful for students with a mathematics disability.

*Other Problems*

Three other characteristics of students who exhibit mathematics disability are the following:

- Problems forming mental representations of mathematical concepts (e.g., a number line, a visual means to represent subtraction as a change process) (Geary 2004)
- Weak ability to access numerical meaning from symbols (i.e., poorly developed number sense) (Gersten and Chard 1999; Rousselle and Noel 2006)
- Problems keeping information in working memory (Passolunghi and Siegel 2004; Swanson and Beebe-Frankenberger 2004)

Theorists disagreed about which, if any, of the characteristics above are fundamental problems for students with difficulties in mathematics. Many studies showed that struggling students had problems in developing and maintaining mental images or representations of such fundamental arithmetic concepts as the base-ten system (Geary 2005). Teachers typically assumed that by third grade, students can easily visualize the base-ten system, and yet some students continue to find this difficult. Students who have a poor working memory often have a hard time performing problems requiring multiple-step solutions (Swanson and Beebe-Frankenberger 2004). Some researchers (Witzel, Mercer, and Miller 2003) have designed interventions that help these students understand numerical concepts by slowly and systematically moving from the concrete to the more abstract, from visual representations of a problem to the world of purely symbolic arithmetic. The influence of the theories of mathematics disabilities and the design of intervention research have been, at best, indirect and informal.

*By Russell Gersten and Benjamin S. Clarke*

**Judith Reed, Series Editor**

#### REFERENCES

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