# American Institute of Mathematical Sciences

February  2022, 2(1): 47-58. doi: 10.3934/steme.2022003

## Conceptual knowledge in area measurement for primary school students: A systematic review

 Department of Mathematics and Sciences Education, Faculty of Education, Universiti Malaya, 50603 Kuala Lumpur, Malaysia; idrus.hafiz89@gmail.com (H.I.); suzieleez@um.edu.my (S.S.A.R.); hutkemri@um.edu.my (H.Z.)

* Correspondence: Email: suzieleez@um.edu.my

Received  January 2022 Revised  February 2022 Published  March 2022

Discussions about teaching area measurement in primary school have been ongoing over some decades. However, investigations that thoroughly examine the current research on conceptual understanding in area measuring in elementary schools are still lacking. The objective of this paper is to review whether conceptual knowledge in area measurement may support students to obtain better results in primary schools. This study is to gain insight into how conceptual knowledge in area measurement has been portrayed for primary school students, and reveal possible omissions and gaps in the synthesized literature on the subject. To gather information, two databases were used: Scopus and Web of Science. Primary searches pulled up many studies on the subject of investigation. After analyzing abstracts and eliminating duplicates, our systematic review indicates that there seems a direct link between conceptual understanding and area measurement in primary school mathematics. Hence, teaching children the principle of area measurement rather than a procedure for solving problems seems to be the most effective way of improving problem-solving skills and conceptual understanding for primary students.

Citation: Hafiz Idrus, Suzieleez Syrene Abdul Rahim, Hutkemri Zulnaidi. Conceptual knowledge in area measurement for primary school students: A systematic review. STEM Education, 2022, 2 (1) : 47-58. doi: 10.3934/steme.2022003
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Publications included and excluded from the systematic review
Analysis of conceptual knowledge by paper
 Definition Discussion Example Grasp for connection Within a domain, relationships "…knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information." [10] Rich in relationship Underlying connections may be few and superficial, or they may be many and deep. "…the quality of one's knowledge of concepts-particularly the richness of the connections inherent in such knowledge" [27] Static knowledge Added data that problem solvers add to the problem and utilize to solve it "…knowledge about facts, concepts, and principles that apply within a certain domain" "…quality characteristic, being the opposite of compiled knowledge" [15] Knowing what semantic networks, hierarchies, and mental models are among the constructions used to describe it. "…consists of the core concepts for a domain and their interrelations" [2] Knowledge of symbol the meanings of symbols "…awareness of what mathematical symbols mean, and the ability to represent relations among numbers in multiple ways."[22] Knowledge as abstract or generic idea covers both general and procedural concepts. Does not have to be verbalized and might be implicit or explicit. "…an abstract or generic idea generalized from particular instances" [23] Underlying structures crucial in the development of procedural knowledge, or the how-to of problem solving, since it helps youngsters to find new techniques and modify existing ones when addressing problems "…understanding of the underlying structures of mathematics" [25] Explicit or Implicit Understanding Higher procedural expertise is associated with greater conceptual understanding. Procedural ability comes before conceptual comprehension. Increased procedural expertise may be achieved through both conceptual and procedure training. Increasing conceptual understanding led to the formation of procedures. "…explicit or implicit understanding of the principles that govern a domain and of the interrelations between pieces of knowledge in a domain" [24] concepts that define an area and are generalized from specific cases A variety of phases are used to build concepts; enactive, iconic & symbolic "…enable us to classify phenomena as belonging, or not belonging, together in certain categories" [13]
 Definition Discussion Example Grasp for connection Within a domain, relationships "…knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information." [10] Rich in relationship Underlying connections may be few and superficial, or they may be many and deep. "…the quality of one's knowledge of concepts-particularly the richness of the connections inherent in such knowledge" [27] Static knowledge Added data that problem solvers add to the problem and utilize to solve it "…knowledge about facts, concepts, and principles that apply within a certain domain" "…quality characteristic, being the opposite of compiled knowledge" [15] Knowing what semantic networks, hierarchies, and mental models are among the constructions used to describe it. "…consists of the core concepts for a domain and their interrelations" [2] Knowledge of symbol the meanings of symbols "…awareness of what mathematical symbols mean, and the ability to represent relations among numbers in multiple ways."[22] Knowledge as abstract or generic idea covers both general and procedural concepts. Does not have to be verbalized and might be implicit or explicit. "…an abstract or generic idea generalized from particular instances" [23] Underlying structures crucial in the development of procedural knowledge, or the how-to of problem solving, since it helps youngsters to find new techniques and modify existing ones when addressing problems "…understanding of the underlying structures of mathematics" [25] Explicit or Implicit Understanding Higher procedural expertise is associated with greater conceptual understanding. Procedural ability comes before conceptual comprehension. Increased procedural expertise may be achieved through both conceptual and procedure training. Increasing conceptual understanding led to the formation of procedures. "…explicit or implicit understanding of the principles that govern a domain and of the interrelations between pieces of knowledge in a domain" [24] concepts that define an area and are generalized from specific cases A variety of phases are used to build concepts; enactive, iconic & symbolic "…enable us to classify phenomena as belonging, or not belonging, together in certain categories" [13]
Analysis of area measurement concept by paper
 Concept Discussion Examples The area of a rectangle with sides l1 and l2 units is given by the formula A = l1 l2 square units Measured in square units, m2, cm2 or mm2 "…The area of a plane figure is the quantity of the plane surface which is enclosed by the perimeter" [21] A tiling of the plane with congruent regions that become units of measure. Acquisition of shapes, measure, computation of measure. Conceptualizing the row-by-column structure of a rectangular array. Arrangement of columns and rows and meaningfully enumerate arrays of a square by using multiplication Knowledge is a more complex subject-matter domain that includes the previous idea of area as well as measurement skills. "…the amount of a 2-D region within a boundary, while area measurement concerns measuring the quantity of a surface enclosed within a 2-D region" [12] A suitable 2D region is chosen as a unit Congruent regions have equal areas Regions do not overlap, the area of the union of two regions is the sum of their areas   • Partitioning,   • Unit Iteration,   • Conservation,   • Structuring an array To generate a two-dimensional measure, measure the lengths of two sides and multiply these one-dimensional units. An array of units is created by iterating a unit along a rectangular area. Requires multiplicative thinking regarding the product of two lengths. "…two-dimensional surface that is contained within a boundary and that can be quantified in some manner" [28] To create a conceptual knowledge of area and perimeter in a concrete way. Not adequately covered in the lower grades, when learners merely learn to define area as the product of length and breadth (A = l × b), which is completely divorced from the idea of covering surface. Learners need objects or resources like bricks and cuttings which they can fit, fold, match and count. "…the amount of surface of a region" [17] Learners need objects or resources like bricks and cuttings which they can fit, fold, match and count. Conserving area as quantity  • Understanding area units  • Structuring rectangular space into composite units  • Understanding area formula  • Distinguish area and perimeter Moving away from physical instruments and toward numerical computing. • In the measuring of teaching and learning, there is a transition. • This course serves as a foundation for more advanced mathematics. • The numerical magnitude of area measurements changes according to the unit's size. "…the quantity of two-dimensional (2D) space enclosed in shapes with closed boundaries, whether they lie on a plane or non- planar surface. "[14] partitioned into equal parts; area measures are the number of area units that fill the space • Units of measurement • A measurement system • Suitable formulas If they do not understand the concept of area, measuring the area of an item might be challenging. "…Area measurement is based on partitioning a region into equally sized units which completely cover it without gaps or overlaps" [32] • Transitivity • The relation between number and measurement • Unit iteration • Operate in area measurement similar to length measurement • Understanding of the attribute area • Equal partitioning • Spatial structuring Conceptual development demand builds to thinking square unit in a row times the number of rows. "…Understanding of area measurement involves learning and coordinating many ideas." [7] Poor performance • Don't completely get the relationship between multiplication and addition • The rectangular array's structure is not readily clear to children. • Inadequate understanding of area and area measurement • Poor stage refers to a proclivity for learning the area formula by role. • Difficulty in generalizing the procedures they have learned "…involves the coordination of two dimensions." [20] Teachers do not give students enough time to learn about the multiplicative structure of rectangular arrays. Start with informal knowledge to Ends with formal knowledge within cognitive plateaus (instruction) What pupils can and cannot accomplish, their conceptualizations and reasoning, cognitive barriers that hinder learning development, and mental processes required for both operating at a level and moving to higher ones "…To construct new knowledge and make sense of novel situations, students build on and revise their current mental structures through the processes of action, reflection, and abstraction." [1]
 Concept Discussion Examples The area of a rectangle with sides l1 and l2 units is given by the formula A = l1 l2 square units Measured in square units, m2, cm2 or mm2 "…The area of a plane figure is the quantity of the plane surface which is enclosed by the perimeter" [21] A tiling of the plane with congruent regions that become units of measure. Acquisition of shapes, measure, computation of measure. Conceptualizing the row-by-column structure of a rectangular array. Arrangement of columns and rows and meaningfully enumerate arrays of a square by using multiplication Knowledge is a more complex subject-matter domain that includes the previous idea of area as well as measurement skills. "…the amount of a 2-D region within a boundary, while area measurement concerns measuring the quantity of a surface enclosed within a 2-D region" [12] A suitable 2D region is chosen as a unit Congruent regions have equal areas Regions do not overlap, the area of the union of two regions is the sum of their areas   • Partitioning,   • Unit Iteration,   • Conservation,   • Structuring an array To generate a two-dimensional measure, measure the lengths of two sides and multiply these one-dimensional units. An array of units is created by iterating a unit along a rectangular area. Requires multiplicative thinking regarding the product of two lengths. "…two-dimensional surface that is contained within a boundary and that can be quantified in some manner" [28] To create a conceptual knowledge of area and perimeter in a concrete way. Not adequately covered in the lower grades, when learners merely learn to define area as the product of length and breadth (A = l × b), which is completely divorced from the idea of covering surface. Learners need objects or resources like bricks and cuttings which they can fit, fold, match and count. "…the amount of surface of a region" [17] Learners need objects or resources like bricks and cuttings which they can fit, fold, match and count. Conserving area as quantity  • Understanding area units  • Structuring rectangular space into composite units  • Understanding area formula  • Distinguish area and perimeter Moving away from physical instruments and toward numerical computing. • In the measuring of teaching and learning, there is a transition. • This course serves as a foundation for more advanced mathematics. • The numerical magnitude of area measurements changes according to the unit's size. "…the quantity of two-dimensional (2D) space enclosed in shapes with closed boundaries, whether they lie on a plane or non- planar surface. "[14] partitioned into equal parts; area measures are the number of area units that fill the space • Units of measurement • A measurement system • Suitable formulas If they do not understand the concept of area, measuring the area of an item might be challenging. "…Area measurement is based on partitioning a region into equally sized units which completely cover it without gaps or overlaps" [32] • Transitivity • The relation between number and measurement • Unit iteration • Operate in area measurement similar to length measurement • Understanding of the attribute area • Equal partitioning • Spatial structuring Conceptual development demand builds to thinking square unit in a row times the number of rows. "…Understanding of area measurement involves learning and coordinating many ideas." [7] Poor performance • Don't completely get the relationship between multiplication and addition • The rectangular array's structure is not readily clear to children. • Inadequate understanding of area and area measurement • Poor stage refers to a proclivity for learning the area formula by role. • Difficulty in generalizing the procedures they have learned "…involves the coordination of two dimensions." [20] Teachers do not give students enough time to learn about the multiplicative structure of rectangular arrays. Start with informal knowledge to Ends with formal knowledge within cognitive plateaus (instruction) What pupils can and cannot accomplish, their conceptualizations and reasoning, cognitive barriers that hinder learning development, and mental processes required for both operating at a level and moving to higher ones "…To construct new knowledge and make sense of novel situations, students build on and revise their current mental structures through the processes of action, reflection, and abstraction." [1]
Levels of comprehend area
 Level Descriptions 1 There is no usage of a composite unit consisting of a row or column of squares (a "line" of squares thought of as a group). At this level, students have trouble visualizing the position of squares in an array and counting square tiles that cover the inside of a rectangle. 2 Partially structured rows or columns. Some pupils, for example, only construct two rows. 3a A set of rows-or column-composites is used to structure an array. At this level, students view the rectangle as being covered by copies of composite units (rows or columns), but they are unable to correlate them with the other dimension. 3b Iteration in a visual row or column. If they can see the rows, these pupils can iterate them (for example, count by fours). 3c Iteration in the inside of a row or column. These kids may use the number of squares in a column to iterate a row. The traditional "formula" approach of estimating area will only have a strong conceptual basis for most pupils at this level. Adapted from Battista et al., [1]
 Level Descriptions 1 There is no usage of a composite unit consisting of a row or column of squares (a "line" of squares thought of as a group). At this level, students have trouble visualizing the position of squares in an array and counting square tiles that cover the inside of a rectangle. 2 Partially structured rows or columns. Some pupils, for example, only construct two rows. 3a A set of rows-or column-composites is used to structure an array. At this level, students view the rectangle as being covered by copies of composite units (rows or columns), but they are unable to correlate them with the other dimension. 3b Iteration in a visual row or column. If they can see the rows, these pupils can iterate them (for example, count by fours). 3c Iteration in the inside of a row or column. These kids may use the number of squares in a column to iterate a row. The traditional "formula" approach of estimating area will only have a strong conceptual basis for most pupils at this level. Adapted from Battista et al., [1]
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