Conceptual Bases of Arithmetic Errors: The Case of Decimal Fractions

Abstract

This study examines children's efforts to make sense of new mathematics instruction. The study documents major categories of errors that appear consistently as children learn decimal fractions. It then establishes the conceptual sources of these errors. Whole number errors derive from children's applying rules for interpreting multidigit integers. Fraction errors derive from children's efforts to interpret decimals as fractions. Different curriculum sequences influence the probability that these classes of errors will appear. It is suggested that errors are a natural concomitant of students' attempts to integrate new material that they are taught with already established knowledge. Since errorful rules cannot be avoided in instruction, educators are encouraged to use them as useful diagnostic tools to detect the nature of children's understanding of a mathematics topic.

Connections Between Teachers’ Knowledge of Students, Instruction, and Achievement Outcomes

“…For instance, one item asked, “Which of the following numbers is greater than 0.23 but less than 0.57?” then posed 0.046 as one of the answer choices. Previous research (Resnick et al, 1989) has established that students know from their study of whole numbers that including a zero to the left of a number does not change the value; overgeneralization of this rule to decimal numbers thus causes errors. This was in fact the case on our assessment as well, where 71% of fourth graders and 61% of fifth graders chose this incorrect option.…”

Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People

“…However, several inherent sources of difficulty interfere with understanding it. The largest is the meaning of the decimal point, which many students fail to understand (Resnick et al, 1989). Whole numbers also do not ordinarily include 0s to the left of the first nonzero digit, but decimals often do (we write .0029 but not ordinarily 0029).…”

University students' misconceptions about rational numbers: Implications for developmental mathematics and instruction of younger students

“…Researchers have also worked to understand the misconceptions students hold with decimals, especially related to foundational understanding. One error identified with fractions also noted with decimals is whole number bias (i.e., interpreting decimals with whole number knowledge; Resnick et al, 1989). This is also referred to as natural number bias (McMullen et al, 2015).…”