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We’ve Got You Covered: The Notion of Area as a Covering

Sandra Davis Trowell

Sandra Davis Trowell is an Associate Professor of Mathematics Education in the Department of Mathematics at Valdosta State University. Her primary responsibilities are to teach mathematics content courses and mathematics education courses for prospective teachers. Her interest include teaching and learning of proportional reasoning and measurement, as well as negotiating rich mathematics classrooms.


Spatial reasoning or visual mathematics is important, even imperative, for students to develop their ideas in mathematics. If mathematics is seen as more than memorization of rules and procedures, then students must have opportunities for sense-making. Principles to Actions (NCTM, 2014) states that “Too much focus is on learning procedures without any connection to meaning, understanding, or the applications that require these procedures.” (p. 3) Measurement is an area of mathematics that permeates all grade levels and most certainly lends itself to spatial reasoning and visual mathematics. If students are to have a conceptual understanding by “seeing” area, then students need to experience and develop a spatial awareness of these ideas. Many students and adults when asked to find the area will ask what formula they need to use. This also implies that the formula for all figures would be the same.

The problems suggested below are intended to be discussed as the class progresses through each question using a problem-centered approach (Wheatley, 1991). It is not appropriate for a teacher to simply model these problems with an assumption that the students will necessarily understand and interpret a solution in the same way that the teacher "sees" a solution. Learners need opportunities to solve and experience these tasks in ways that are meaningful to them. They need opportunities to consider the tasks, have discussions in small groups or pairs, and then move to whole-class discussions. The whole class discussions should revolve around the feasibility and viability of the students' solutions and ideas. The focus should be upon sense-making, communication, and making connections.

Presenting problems that have the potential to develop area ideas begin with tasks that involve building rectangles with squares. This makes sense as the area is defined in terms of square units and this provides the spatial experience for students to build and begin to visualize the area of rectangles. Students should be given square tiles to consider these tasks with the problems progressing so that students do not have enough squares to “cover” the rectangle. These tiles may be commercially produced tiles or tiles cut from paper. The covering process is an important action for the student to take and not simply witness.

First give students a sheet with a rectangle outlined, choosing a rectangle whose dimensions will fit on a sheet that can be covered with the squares that are available to each student:


The students may ask questions about spaces left open, etc. One possible context would be to consider the squares as tiles that will be used to cover the rectangle. Ask students to do this individually and be mindful of how they consider their answers. Follow this by giving them a second rectangle that they can cover with the squares that are available to them. i.e. 2 by 4, 3 by 4, 2 by 5, etc. The goal is for students to begin imaging the covering of the rectangle and generalize what is happening when they need to find the number of squares that would cover the rectangle. Ask the students to share how they are thinking about covering this plane. Students may add the rows, add the columns, or multiply the dimensions. Some may remember playing with rectangular arrays and can connect with that multiplicative thinking. Some may simply count the number of squares at this point. Classroom conversations are important so that students can try to make sense of each other’s ideas and ways of thinking. Through their sharing, the students can have opportunities to defend and elaborate upon their ideas about area as a covering.

After students have had opportunities to build their square coverings, the next step would be to give them rectangles that they cannot cover with the squares they are given. Ask them to work individually to consider how they know how many squares they will need to cover their new rectangles. Possible dimensions would include 3 by 5, 5 by 6, or 7 by 10. The dimensions should not be so large that it gets in the way of the student thinking about how to organize their covering. This is to provide opportunities for the students to build their image or visual for the area of a rectangle. Again, after students have had adequate time to reflect on this problem, it is important to have a whole-class discussion about patterns and generalizations that they notice. Do not expect all students to initially say that the area of a rectangle is length times width, but it will most likely come out of their conversations. Coordination of the rows and columns is not automatic for students and we should not assume that they see what we might see. These conversations allow students to make sense of the area and may prompt them to think about multiples and arrays.

Finding the area of rectangles is a meaningful part of mathematics as many ideas arise from understanding and imaging the area as a covering of square units. From the area of a rectangle, one can then move to ask students to consider parallelograms, triangles, and trapezoids. Finding the area of these polygons can be meaningful if students have opportunities to consider how these relate to what they know about the area of rectangles. Thinking in arrays can be meaningful when students consider multiplication of two-digit numbers. Students cannot simply be shown an area model for this kind of multiplication with the expectation that it will make sense if they had not yet made sense of the area. Taking opportunities to build meaning for area can impact their subsequent mathematical ideas.

Building rectangles without enough squares to cover their rectangles promotes the spatial sense of covering the plane. Asking students to communicate and generalize their ideas and to use multiple representations is consistent with the National Council of Mathematics suggestions. Research recommends that “mathematics teaching and learning need to become more visual – there is not a single idea or concept that cannot be illustrated or thought about visually” (Boaler et al., 2016, p. 329). Students may build richer and more sophisticated mathematical ideas when addressing a few well thought out questions followed by meaningful discussions rather than working through many area exercises in which they are simply plugging numbers in a stated algorithm.

The Teaching and Learning Guiding Principle in Principles for School Mathematics in Principles to Actions(NCTM, 2014) states:

An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically. (p. 5)

The above-mentioned problems focus upon building a sense of area in a mathematics classroom that is consistent with this principle.  If presented in a problem-centered learning environment, conversations can promote rich mathematics.  Allowing time for students to discuss, defend, and explain their mathematics is an essential part of learning, as is listening to and make sense of other's mathematics.  Such tasks as these can be a rich part of an effective mathematics program.


References

Boaler J, Chen L, Williams C, Cordero M (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied and Computational Mathematics, 5(5), pp. 325-330.

National Council of Teachers of Mathematics. (2014). Principles to actions: ensuring mathematical success for all. Reston, VA: NCTM.

Wheatley, G. J. (1991). Constructivist perspectives on mathematics and science learning. Science Education, 15, 9-21.


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