Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation’s Challenges

Abstract

Research has found that elementary students face five main challenges in learning area measurement: (1) conserving area as a quantity, (2) understanding area units, (3) structuring rectangular space into composite units, (4) understanding area formulas, and (5) distinguishing area and perimeter. How well do elementary mathematics curricula address these challenges? A detailed analysis of three U.S. elementary textbook series revealed systematic deficits. Each presented area measurement in strongly procedural terms using a shared sequence of procedures across grades. Key conceptual principles were infrequently expressed and often well after related procedures were introduced. Particularly weak support was given for understanding how the multiplication of lengths produces area measures. The results suggest that the content of written curricula contributes to students' weak learning of area measurement. digitalcommons.unl.edu

How well aligned are common core textbooks to students’ development in area measurement?

“…In the classroom, textbooks offer opportunities to learn (OTL) mathematics (Hong, Choi, Runnalls, & Hwang, ; Otten et al, ; Smith, Males, Dietiker, Lee, & Mosier, ; Smith et al, ). OTL is often defined as whether or not students have had the opportunity to study a particular topic or learn how to solve a particular type of problem (Floden, ; Liu, ).…”

“…United States eighth‐grade students’ performance in measurement was weaker than any other content area on the National Assessment of Educational Progress and the Third International Mathematics and Science Study (Mullis, Martin, Foy, & Arora, ; Thompson & Preston, ), indicating that students may need additional mathematical tasks and instruction that reflect their developmental thinking on the matter. We chose the topic of area measurement in geometry because it is one commonly used domain of measurement in everyday life (Outhred & Mitchelmore, ), it plays a transitional role between length and volume measurements, and it forms a foundation for more advanced mathematics, ranging from fractions to calculus (Smith, Males, & Gonulates, ). Studies have shown that students often rely solely on area formulas, without understanding how or why they work (Baturo & Nason, ; Zacharos, ).…”

“…Studies have shown that students often rely solely on area formulas, without understanding how or why they work (Baturo & Nason, ; Zacharos, ). Furthermore, a recent study indicated that popular U.S. textbooks included only limited number of expositions, worked examples and exercise problems related to important conceptual ideas in area measurement such as covering, partitioning, and array structure (Smith et al, ). Since many previous textbook analysis studies have examined textbooks that were developed prior to the development of CCSSM (Hong & Choi, ; Otten, Gilbertson, Males, & Clark, ; Smith et al, ; Son & Hu, ), examining three Common Core–aligned textbook series will further our understanding of how recent textbooks treat area measurement, as well as expand previous findings by comparing textbook coverage and sequencing to research‐backed learning trajectories for area.…”

Teaching Area Measurement and Subgoals through Problem Solving

“…Therefore, to understand the numerical result of area formulae, students are required to reconstruct their understanding of multiplication as multiplicative composition (Confrey, 2012;Lehrer, 2003;Smith et al, 2016). That is, multiplication of two 1D lengths produces area.…”

“…These objectives state that middle school students should be able to apply area formulae to solve authentic problems, for example, "estimating and working out the area of a vegetable garden in square metres and calculating how much sugarcane mulch to buy to cover it" (ACARA, 2016, p. 17). Second, area measurement has applications in other school disciplines, particularly in science and technology (Huang & Witz, 2011;Smith, Males, & Gonulates, 2016), as well as in vocational subjects. Finally, the reasoning that students develop while learning area measurement can be applied in the learning of other mathematical strands such as operations with numbers, volume, multiplication, fractions, probability, algebraic multiplication, enlargement, and calculus (Davydov, Gorbov, Mukulina, Savelyeva, & Tabachnikova, 1999;Dougherty & Zilliox, 2003;Fernández, De Bock, Verschaffel, & Van Dooren, 2014;Hirstein et al, 1978;Kordaki & Potari, 1998;Outhred & Mitchelmore, 2000;Sarama & Clements, 2009;Tan Sisman & Aksu, 2016).…”