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

Academic Editor: Stephen Xu

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
References:
[1]

Battista, M.T., Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement. Mathematical Thinking and Learning, 2004, 6(2): 185–204. https://doi.org/10.1207/s15327833mtl0602_6 doi: 10.1207/s15327833mtl0602_6.

[2]

Byrnes, J.P. and Wasik, B.A., Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 1991, 27(5): 777–786. https://doi.org/10.1037//0012-1649.27.5.777 doi: 10.1037//0012-1649.27.5.777.

[3]

Castle, K. and Needham, J., (2007). First graders' understanding of measurement. Early Childhood Education Journal, 2007, 35(3): 215‒221. https://doi.org/10.1007/s10643-007-0210-7

[4]

Clements, D.H., et al., Sarama, J., Van Dine, D.W., Barrett, J.E., Cullen, C.J., Hudyma, A., Dolgin, R., Cullen, A. L., & Eames, C. L. (2018b). Evaluation of three interventions teaching area measurement as spatial structuring to young children. Journal of Mathematical Behavior, 2018, 50: 23–41. https://doi.org/10.1016/j.jmathb.2017.12.004

[5]

Common Core State Standards Initiative, Common Core State Standards for Mathematics, 2010. In Development. http://www.corestandards.org/

[6]

Crooks, N.M. and Alibali, M.W., Defining and Measuring Conceptual Knowledge in Mathematics. Developmental Review, 2014, 34(4): 344–377. https://doi.org/10.1016/j.dr.2014.10.001 doi: 10.1016/j.dr.2014.10.001.

[7]

Cross, C.T. and Woods, T.A., Mathematics Learning in Early Childhood Paths Toward Excellence and Equity (H. Schweingruber (ed.)), The National Academic Press, 2009.

[8]

Grewal, A., Kataria, H. and Dhawan, I., Literature search for research planning and identification of research problem. Indian Journal of Anaesthesia, 2016, 60(9): 635–639. https://doi.org/10.4103/0019-5049.190618 doi: 10.4103/0019-5049.190618.

[9]

Hiebert, J., Why Do Some Children Have Trouble Learning Measurement Concepts? The Arithmetic Teacher, 1984, 31(7): 19–24. https://doi.org/10.5951/at.31.7.0019 doi: 10.5951/at.31.7.0019.

[10]

Hiebert, J., Conceptual and procedural knowledge: The case of mathematics. In Conceptual and Procedural Knowledge: The Case of Mathematics. Lawrence Erlbaum Associates, 1986. https://doi.org/10.4324/9780203063538

[11]

Hord, C. and Xin, Y.P., Teaching Area and Volume to Students with Mild Intellectual Disability. Journal of Special Education, 2015, 49(2): 118‒128. https://doi.org/10.1177/0022466914527826 doi: 10.1177/0022466914527826.

[12]

Huang, H.-M.E. and Witz, K.G., Children's Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems. Journal of Curriculum and Teaching, 2013, 2(1): 10–26. https://doi.org/10.5430/jct.v2n1p10 doi: 10.5430/jct.v2n1p10.

[13]

Hurrell, D.P., Conceptual Knowledge OR Procedural Knowledge or Conceptual Knowledge AND Procedural Knowledge: Why the Conjunction is Important to Teachers. Australian Journal of Teacher Education, 2021, 46(2): 57–71. https://doi.org/10.14221/ajte.2021v46n2.4 doi: 10.14221/ajte.2021v46n2.4.

[14]

II, J.P.S., Males, L. and Gonulates, F., Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation's Challenges. Faculty Publications: Department of Teaching, Learning and Teacher Education, 2016, 18(4): 239–270. https://doi.org/10.1080/10986065.2016.1219930 doi: 10.1080/10986065.2016.1219930.

[15]

De Jong, T. and Ferguson-Hessler, M.G.M., (1996). Types of qualities of knowledge. Educational Psychologist, 1996, 31(2), 105–113. https://doi.org/10.1207/s15326985ep3102_2

[16]

Lehrer, C., Developing Understanding of Measurement. A Research Companion to Principles and Standards for School Mathematics, 2003: 179–192.

[17]

Machaba, F.M., The concepts of area and perimeter: Insights and misconceptions of Grade 10 learners. Pythagoras - Journal of the Associantion for Mathematics Education of South Africa, 2016, 37(1): 1–11. https://doi.org/10.4102/pythagoras.v37i1.304 doi: 10.4102/pythagoras.v37i1.304.

[18]

Mendezabal, M.J.N. and Tindowen, D.J.C., Improving Students' Attitude, Conceptual Understanding and Procedural Skills in Differential Calculus Through Microsoft Mathematics. Journal of Technology and Science Education, 2018, 4(4): 385–397. https://doi.org/10.3926/jotse.356 doi: 10.3926/jotse.356.

[19]

Naidoo, N., Creating a Deeper Understanding of Area and Perimeter in the Primary Classroom.

[20]

Outhred, L.N. and Mitchelmore, M.C., Young children's intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 2000, 31(2): 144–167. https://doi.org/10.2307/749749 doi: 10.2307/749749.

[21]

Perry, O. and Perry, J., Mathematics I. In Macmillan Technician Series (1st ed.), 1981. https://doi.org/10.1007/978-1-349-05230-1

[22]

Ploger, D. and Hecht, S., Enhancing children's conceptual understanding of mathematics through chartworld software. Journal of Research in Childhood Education, 2009, 23(3): 267–277. https://doi.org/10.1080/02568540909594660 doi: 10.1080/02568540909594660.

[23]

Rittle-Johnson, B., Iterative development of conceptual and procedural knowledge in mathematics learning and instruction. In The Cambridge Handbook of Cognition and Education, 2019: 124–147. https://doi.org/10.1017/9781108235631.007

[24]

Rittle-Johnson, B. and Alibali, M.W., Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 1999, 91(1): 175–189. https://doi.org/10.1037/0022-0663.91.1.175 doi: 10.1037/0022-0663.91.1.175.

[25]

Robinson, K.M. and Dube, A.K., A microgenetic study of the multiplication and division inversion concept. Canadian Journal of Experimental Psychology, 2009, 63(3): 193–200. https://doi.org/10.1037/a0013908 doi: 10.1037/a0013908.

[26]

Sholihah, S.Z. and Afriansyah, E.A., Analisis Kesulitan Siswa dalam Proses Pemecahan Masalah Geometri Berdasarkan Tahapan Berpikir Van Hiele. Mosharafa: Jurnal Pendidikan Matematika, 2017, 6(2): 287–298. https://doi.org/10.31980/mosharafa.v6i2.317 doi: 10.31980/mosharafa.v6i2.317.

[27]

Star, J.R., Reconceptualizing Procedural Knowledge. Journal for Research in Mathematics Education, 2005, 36(5): 404–411.

[28]

Stephen, M. and Clements, D.H., Linear and Area Measurement in Prekindergarten to Grade 2. Learning and Teaching Measurement, 2003, 5(1): 3‒16.

[29]

Tan Sisman, G. and Aksu, M., A Study on Sixth Grade Students' Misconceptions and Errors in Spatial Measurement: Length, Area, and Volume. International Journal of Science and Mathematics Education, 2015, 14(7): 1293–1319. https://doi.org/10.1007/s10763-015-9642-5 doi: 10.1007/s10763-015-9642-5.

[30]

Tan Şişman, G. and Aksu, M., Sixth grade students' performance on length, area, and volume measurement. Egitim ve Bilim, 2012, 37(166): 141–154.

[31]

Wahid, N.T.A., Talib, O., Sulaiman, T. and Puad, M.H.M., A Systematic Literature Review on the Problem-Posing Strategies for Biology Problem-Posing Multimedia Module Design. International Journal of Academic Research in Business and Social Sciences, 2018, 8(12): 1020–1032. https://doi.org/10.6007/ijarbss/v8-i12/5150 doi: 10.6007/ijarbss/v8-i12/5150.

[32]

Yuberta, K.R., Supporting Students' Understanding of Area Measurement Using Rme Approach. International Conference on Education 2018 Teachers in the Digital Age, 2019, 3(1): 199–206.

show all references

References:
[1]

Battista, M.T., Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement. Mathematical Thinking and Learning, 2004, 6(2): 185–204. https://doi.org/10.1207/s15327833mtl0602_6 doi: 10.1207/s15327833mtl0602_6.

[2]

Byrnes, J.P. and Wasik, B.A., Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 1991, 27(5): 777–786. https://doi.org/10.1037//0012-1649.27.5.777 doi: 10.1037//0012-1649.27.5.777.

[3]

Castle, K. and Needham, J., (2007). First graders' understanding of measurement. Early Childhood Education Journal, 2007, 35(3): 215‒221. https://doi.org/10.1007/s10643-007-0210-7

[4]

Clements, D.H., et al., Sarama, J., Van Dine, D.W., Barrett, J.E., Cullen, C.J., Hudyma, A., Dolgin, R., Cullen, A. L., & Eames, C. L. (2018b). Evaluation of three interventions teaching area measurement as spatial structuring to young children. Journal of Mathematical Behavior, 2018, 50: 23–41. https://doi.org/10.1016/j.jmathb.2017.12.004

[5]

Common Core State Standards Initiative, Common Core State Standards for Mathematics, 2010. In Development. http://www.corestandards.org/

[6]

Crooks, N.M. and Alibali, M.W., Defining and Measuring Conceptual Knowledge in Mathematics. Developmental Review, 2014, 34(4): 344–377. https://doi.org/10.1016/j.dr.2014.10.001 doi: 10.1016/j.dr.2014.10.001.

[7]

Cross, C.T. and Woods, T.A., Mathematics Learning in Early Childhood Paths Toward Excellence and Equity (H. Schweingruber (ed.)), The National Academic Press, 2009.

[8]

Grewal, A., Kataria, H. and Dhawan, I., Literature search for research planning and identification of research problem. Indian Journal of Anaesthesia, 2016, 60(9): 635–639. https://doi.org/10.4103/0019-5049.190618 doi: 10.4103/0019-5049.190618.

[9]

Hiebert, J., Why Do Some Children Have Trouble Learning Measurement Concepts? The Arithmetic Teacher, 1984, 31(7): 19–24. https://doi.org/10.5951/at.31.7.0019 doi: 10.5951/at.31.7.0019.

[10]

Hiebert, J., Conceptual and procedural knowledge: The case of mathematics. In Conceptual and Procedural Knowledge: The Case of Mathematics. Lawrence Erlbaum Associates, 1986. https://doi.org/10.4324/9780203063538

[11]

Hord, C. and Xin, Y.P., Teaching Area and Volume to Students with Mild Intellectual Disability. Journal of Special Education, 2015, 49(2): 118‒128. https://doi.org/10.1177/0022466914527826 doi: 10.1177/0022466914527826.

[12]

Huang, H.-M.E. and Witz, K.G., Children's Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems. Journal of Curriculum and Teaching, 2013, 2(1): 10–26. https://doi.org/10.5430/jct.v2n1p10 doi: 10.5430/jct.v2n1p10.

[13]

Hurrell, D.P., Conceptual Knowledge OR Procedural Knowledge or Conceptual Knowledge AND Procedural Knowledge: Why the Conjunction is Important to Teachers. Australian Journal of Teacher Education, 2021, 46(2): 57–71. https://doi.org/10.14221/ajte.2021v46n2.4 doi: 10.14221/ajte.2021v46n2.4.

[14]

II, J.P.S., Males, L. and Gonulates, F., Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation's Challenges. Faculty Publications: Department of Teaching, Learning and Teacher Education, 2016, 18(4): 239–270. https://doi.org/10.1080/10986065.2016.1219930 doi: 10.1080/10986065.2016.1219930.

[15]

De Jong, T. and Ferguson-Hessler, M.G.M., (1996). Types of qualities of knowledge. Educational Psychologist, 1996, 31(2), 105–113. https://doi.org/10.1207/s15326985ep3102_2

[16]

Lehrer, C., Developing Understanding of Measurement. A Research Companion to Principles and Standards for School Mathematics, 2003: 179–192.

[17]

Machaba, F.M., The concepts of area and perimeter: Insights and misconceptions of Grade 10 learners. Pythagoras - Journal of the Associantion for Mathematics Education of South Africa, 2016, 37(1): 1–11. https://doi.org/10.4102/pythagoras.v37i1.304 doi: 10.4102/pythagoras.v37i1.304.

[18]

Mendezabal, M.J.N. and Tindowen, D.J.C., Improving Students' Attitude, Conceptual Understanding and Procedural Skills in Differential Calculus Through Microsoft Mathematics. Journal of Technology and Science Education, 2018, 4(4): 385–397. https://doi.org/10.3926/jotse.356 doi: 10.3926/jotse.356.

[19]

Naidoo, N., Creating a Deeper Understanding of Area and Perimeter in the Primary Classroom.

[20]

Outhred, L.N. and Mitchelmore, M.C., Young children's intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 2000, 31(2): 144–167. https://doi.org/10.2307/749749 doi: 10.2307/749749.

[21]

Perry, O. and Perry, J., Mathematics I. In Macmillan Technician Series (1st ed.), 1981. https://doi.org/10.1007/978-1-349-05230-1

[22]

Ploger, D. and Hecht, S., Enhancing children's conceptual understanding of mathematics through chartworld software. Journal of Research in Childhood Education, 2009, 23(3): 267–277. https://doi.org/10.1080/02568540909594660 doi: 10.1080/02568540909594660.

[23]

Rittle-Johnson, B., Iterative development of conceptual and procedural knowledge in mathematics learning and instruction. In The Cambridge Handbook of Cognition and Education, 2019: 124–147. https://doi.org/10.1017/9781108235631.007

[24]

Rittle-Johnson, B. and Alibali, M.W., Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 1999, 91(1): 175–189. https://doi.org/10.1037/0022-0663.91.1.175 doi: 10.1037/0022-0663.91.1.175.

[25]

Robinson, K.M. and Dube, A.K., A microgenetic study of the multiplication and division inversion concept. Canadian Journal of Experimental Psychology, 2009, 63(3): 193–200. https://doi.org/10.1037/a0013908 doi: 10.1037/a0013908.

[26]

Sholihah, S.Z. and Afriansyah, E.A., Analisis Kesulitan Siswa dalam Proses Pemecahan Masalah Geometri Berdasarkan Tahapan Berpikir Van Hiele. Mosharafa: Jurnal Pendidikan Matematika, 2017, 6(2): 287–298. https://doi.org/10.31980/mosharafa.v6i2.317 doi: 10.31980/mosharafa.v6i2.317.

[27]

Star, J.R., Reconceptualizing Procedural Knowledge. Journal for Research in Mathematics Education, 2005, 36(5): 404–411.

[28]

Stephen, M. and Clements, D.H., Linear and Area Measurement in Prekindergarten to Grade 2. Learning and Teaching Measurement, 2003, 5(1): 3‒16.

[29]

Tan Sisman, G. and Aksu, M., A Study on Sixth Grade Students' Misconceptions and Errors in Spatial Measurement: Length, Area, and Volume. International Journal of Science and Mathematics Education, 2015, 14(7): 1293–1319. https://doi.org/10.1007/s10763-015-9642-5 doi: 10.1007/s10763-015-9642-5.

[30]

Tan Şişman, G. and Aksu, M., Sixth grade students' performance on length, area, and volume measurement. Egitim ve Bilim, 2012, 37(166): 141–154.

[31]

Wahid, N.T.A., Talib, O., Sulaiman, T. and Puad, M.H.M., A Systematic Literature Review on the Problem-Posing Strategies for Biology Problem-Posing Multimedia Module Design. International Journal of Academic Research in Business and Social Sciences, 2018, 8(12): 1020–1032. https://doi.org/10.6007/ijarbss/v8-i12/5150 doi: 10.6007/ijarbss/v8-i12/5150.

[32]

Yuberta, K.R., Supporting Students' Understanding of Area Measurement Using Rme Approach. International Conference on Education 2018 Teachers in the Digital Age, 2019, 3(1): 199–206.

Figure 1.  Publications included and excluded from the systematic review
Table 1.  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]
Table 2.  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]
Table 3.  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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