One difference between top-performing math students and other average and even high-achieving math students, says a recent study in School Science and Mathematics, is that top-performing students know that ¼ has more than one meaning. They know that it can represent a part-whole relationship but also a quotient, a ratio, a measure or an operation. Average and high-achieving peers, on the other hand, are more likely to see ¼ as representing a part-whole relationship only, write the researchers of this study of 4th-grade students.
“Discerning the differences among representations is crucial to competent reasoning with differing rational number relations,” write the authors. “However, mathematics curricula and teaching frequently fail to emphasize making connections across various rational number relations.”
Lack of flexible understanding
Average and above-average students are not developing a flexible understanding of the multiple meanings of rational numbers based on the results of this study of 4th-graders, write the authors. When 52 students were asked to place task cards into “mathematical categories that show which ones you think are mathematically related,” most (58%) grouped the cards based on surface characteristics, they report. They grouped the cards based on whether they had words, numbers or pictures rather than on underlying mathematical similarities, the researchers write.
Most students even failed to group cards into a part-whole relationship category. Top-performing students produced many more groupings based on underlying quantity meanings. There are 5 families of rational number relations or perspectives, the authors write:
- Part-whole relationship
- operator (e.g. a number used to proportionally expand or reduce)
“Overall, the students’ groupings show few viable conceptions about what made rational number representations mathematically similar,” the authors write.
“Instead, the patterns illustrate that a strong and largely uniform tendency exists for students to cluster word problems, numerical notations, and visual displays without respect to the underlying quantity they convey.”
Participants were from 3 schools in the southeastern part of the U.S. that were selected based on state testing results. The schools had high percentages of native speakers of English. Two of the schools were high-performing schools and one was an average-performing school. Top students from the two top-performing schools and a typical sample of students from the 3rd school were selected to participate in the study. Of 91 students invited to participate, 52 obtained parental consent to be included.
Teacher knowledge and curricula could be factors in the difference between the top students and other students, authors speculate. Unfortunately, teachers themselves may possess a weak understanding of rational numbers, they write. Also two different curricula were used in the schools: Two groups of students were taught with Math Advantage while the top students were being taught with Everyday Mathematics.
“One key difference between the two textbooks is that the reform textbook (Everyday Mathematics) exposes students to a wide range of rational number representations whereas the traditional Textbook (Math Advantage) concentrates on showing visual representations of the part-whole relation,” they write.
Students should be provided with opportunities to explore and develop an understanding of multiple perspectives and representations of rational numbers, they conclude, because this is of crucial importance in their future performance in mathematics.
“Identifying Fourth Graders’ Understanding of Rational Number Representations: A Mixed Methods Approach,” by Bryan Mosely and Yukari Okamoto, School Science and Mathematics, Volume 108, Number 6, pp. 238-250.