Abstract:

Mathematical skills are acquired through a cumulative and progressive learning trajectory. Numeracy or number knowledge (i.e., the understanding of whole numbers and number relationships) has been identified as a predictor of successful mathematics learning (Fazio, Bailey, Thompson, & Siegler, 2014). Most children are able to acquire number knowledge skills when taught informally (e.g., home) or formally (e.g., school). However, not all children acquire numeracy skills easily. Theories of number competency have been linked to several factors such as family income, working memory capacity, and language abilities (Nys, Content, & Leybaert, 2013; Passolunghi & Lanfranchi, 2012). Researchers have found that students who have mathematics learning disability are less efficient and slower in processing symbolic number tasks such as magnitude comparison tasks than typically developing children (De Smedt & Gilmore, 2011).

Recognizing that persistent errors in number knowledge tasks may imply incorrect or incomplete number concepts, we conducted a study in which we examined the type of errors Portuguese students made in three number knowledge tasks.

The purposes of our study were to:
1) examine the type of errors Portuguese elementary students make in number knowledge tasks,
2) identify the most common type of errors in number knowledge by grade level, and
3) identify the type of errors that significantly contribute to student performance on number knowledge.

In the present study, we examined number knowledge skills of 697 students from first to fourth grade, from seven elementary schools in Portugal. Students completed three number knowledge tasks:
1) translating numbers into words,
2) symbolic magnitude (i.e., number comparison), and
3) decomposing numbers.

Participants were 697 students, grades first through fourth, from seven elementary schools in Portugal. Students completed a written assessment test of mathematical knowledge developed by Lopes and Bueno (2014). The instrument reflects the Portuguese mathematics curriculum of 1st through 4th grade. We evaluated students’ answers by means of error analysis using a three-category coding system adapted from specific error types that were computed by grade level.

The three categories were:
1) prerequisite linguistic (e.g., write digits according to value),
2) vocabulary (e.g., two, one hundred one for 2,101), and
3) compound construction (e.g., inconsistent syntactic structure [one thousand hundred for 100,000]).

Data frequency analyses showed that the most common type of errors among all students were the prerequisite linguistic type and the compound construction type. Results showed that there were significant differences among grades and that the prerequisite linguistic error type, particularly in the magnitude tasks, significantly contributed to students’ performance on number knowledge tasks. This is important for instruction because learning difficulties in mathematics have been associated with weaknesses in intermediate number knowledge competencies. Our qualitative data analyses suggest that instruction need to consider instructing students the pre linguistic rules of the number system, specifically the principles related to larger numbers. In sum, this study highlights the critical need for explicit and effective instruction of numeracy skills and concepts to students in both the lower and upper elementary grades.

Keywords:

Mathematics, number knowledge, error analysis, intervention.