Visual Discrimination: Spatial Reasoning Activity for Enhancing Children’s Spatial Skills

Abstract

Children can connect with and grasp complex geometric concepts when we harness and integrate spatial thinking into learning situations. In this chapter, we’ll look at a couple warm-ups and a few classroom exercises that show how thoughtful resources and unique geometry assignments may help students enhance their understanding in the complex context of the K-12 classroom. Throughout the class, students used a number of spatial abilities, such as visualization and mental rotations, while building and working with polyominoes.

1. Introduction

A person’s capacity to display and manage spatial figures, relationships, and figure formations are referred to as their “spatial reasoning aptitude” [1, 2]. In other words, when it comes to thinking spatially, we think about the placement and movement of things and ourselves, both mentally and physically. There are many ideas, tools, and methods that make up this skill or process [3]. Mathematical exploration and solutions, especially non-routine ones, depend largely on spatial thinking, and many students find mathematics easier to learn when it is handled spatially [4].

Although it is seldom that spatial thinking is explicitly taught in schools, spatial vocabulary and principles like rotation, visualization, identification, and translation are indeed often employed in math disciplines like geometry and calculus. Researchers have developed a strategy for providing explicit spatial instruction that involves collaborating with classroom instructors to create spatial interventions [5, 6]. Such instruction typically focuses on developing selected aspects of spatial reasoning skills through the use of up-to-date methods and processes that are helpful for solving nuanced math classroom problems. There is a need to distinguish between spatial ability and spatial skills. Spatial ability is described as a person’s natural capacity to envision a solution before having received any formal instruction, i.e., a person is born with the ability. In contrast, spatial skills are learned or gained via training. Tartre [7] distinguishes two types of spatial skills: spatial visualization and spatial orientation. Spatial visualization is the art of visualizing 2- and 3-dimensional objects in one’s mind. At rest and in motion, spatial orientation is the capacity to maintain our bodily orientation and posture concerning the surrounding environment (physical space) [8]. According to research, spatial thinking skills and geometric reasoning are essential to the development of problem-solving abilities, mathematics learning, and reading [9, 10]. Strengthening the link between spatial thinking and mathematical accomplishment. Mathematical knowledge improves as spatial thinking improves since both can be learned and improved through practice [11, 12, 13, 14, 15]. This study will explore activities that help students enhance their spatial skills.

2. Research method

The subjects of this qualitative case study were 40 middle school students from low-income neighborhoods in one of the southern states of the United States. Throughout the 2017–2018 school year, students worked on honing their spatial reasoning abilities by using manipulatives. Data was collected using various methods, including interviews, observations, artifacts, and video recordings. Data was initially input into a word processor before being entered into NVivo (qualitative data analysis software) to be coded and evaluated.

3. Results

3.1 The warm-up activity

The teacher started the class with a warm-up activity involving visual discrimination, which is a visual perception skill that refers to the ability to differentiate one object from another. The development of visual discrimination skills can help a person to compare and contrast visual images accurately and can likewise enhance one’s ability to think and see things differently. The capacity to visually identify letters and words becomes vital in learning to read; visual discrimination must essentially occur at all times while a person is reading [16]. One must be able to discriminate visually in terms of color, foreground-background, form, size, and position in space. Observation is a fundamental thinking skill as it underlies and supports other identification skills for gathering information.

The warm-up activity was called “I’m going on a hike.” For this activity, students were shown on the overhead projector a picture of a jungle with twigs, leaves, trees, and a snake that required careful observation to identify. Some students were able to locate the snake, whereas others did not see it until the teacher traced it out.

Teacher: Alright, visual discrimination, I’m going on a hike. I’m going on a hike. Do you see anything?

Student 1: Grass.

Student 2: Grass, leaves, twigs.

Teacher: I see a tree. I see branches. I see dirt.

Student 3: I see a cobra head

Teacher: You see a cobra head? Come forward and show me.

Student 3: I lost it.

Teacher: Maybe under the leaves?

Student 4: I see it. It looks like a mop, though. You don’t see… It’s right there.

Student 4: Oh, I see it. I might get excited if I see that when I go on a hike.

The students used visual discrimination to find the snake, which was not apparent to some students.

The teacher also gave another warm-up about going on a picnic.

Teacher: Okay, the game we’re going to play is called, “I’m going on a picnic.” You happen to know it; don’t give any hints out, okay? We’re going to see who can figure this out. Alright, are you all ready to get started?

Students: Yeah [nodding].

Teacher: Alright, here it goes [crosses her arms]. I’m going on a picnic, and I’m going to bring… chocolate chips. Good job, Ms. Hanes, you can go. Okay [calls on Student 38].

Student 38: I’m going on a picnic, and I’m bringing apple pie?

Teacher: You want to bring apple pie. Well, come on up here with me [Student 38 goes to stand next to the teacher in front of the class].

Student 39: I’m going on a picnic, and I want to bring pizza.

Teacher: Can she come [asks Student 38]?

Student 38: No [shakes his head].

Teacher: Nope. Uh, I hope you can come to our thing too, or you… I think you might not be doing the right thing [to Student 38]. Oops, did you forget [to Student 38]? You have to sit back down. You’re not remembering [sends Student 38 back to his seat]. Alright, yes, ma’am [to Student 40].

Student 40: I’m going on a picnic, and I’m going to bring blueberries.

Teacher: Oh, I’m sorry, I don’t need blueberries. Yes [to Student 41].

Student 41: I’m going on a picnic, and I’m bringing an elephant.

Teacher: An elephant, I would love an elephant; come on up here![Other students in the class laugh.] Okay, yes, ma’am [to Student 42].

Student 42: I’m going on a picnic, and I’m bringing sandwiches. [At the same time Student 42 is speaking, Student 43 is also saying the following].

Student 43: I’m going on a picnic, and I’m bringing watermelon.

Teacher: Uh, watermelon and sandwiches, you said watermelon, right? Okay, both of you all come up. The purpose of the exercise was for students to emulate the teacher’s body language while declaring in front of the class what they would bring to qualify for the picnic. It did not matter what one brought to the picnic; what was important was how well the students emulated the teacher’s body language.

For example, the instructor crossed her arms while discussing what to bring to the picnic and expected pupils to do the same. Other physical signals utilized were eye closure, making a fist, and standing on one leg. Many students did not qualify for going on a picnic as they did not visually discriminate what was going on with the teacher’s body language. The students loved the game and engaged in what was going on. The teacher transitioned to the next activity, which involved distributing one domino to each student.

3.2 Dominoes

Teacher: Okay, now that is awesome. We’re going to be doing many things here that have to do with visual [points to her eyes]. So, when you walk in here, make sure your eyes are open because they will do great stuff. Now, since we just talked about visualization and I’m going to give everybody one of these things [holds up bucket of dominoes and starts passing out dominoes to the students]. You all know what these things are?

Students: No, yeah, dominoes! (Figure 1)

Figure 1.

Dominoes.

Teacher: This is your vision at the moment, so pay attention to it. What did you observe about these dominos?

Student 4: There’s a dragon on the back.

Teacher: Alright, cool, so there’s an animal on the back. What else did you notice [points to Student 5]?

Student 5: They’re black.

Teacher: What else did you notice [points to Student 6]?

Student 6: White dots.

Teacher: What did you see [points to Student 8]?

Student 8: They have a middle line separate.

Teacher: Ooh, a line, 2 sections. Ooh, I want to work on that little line thing; if you all can think of more things about her line, that’s a good start there.

Student 10: Have two sections.

Teacher: Ooh, two sections; kind of goes with her idea, thing. Tell me about those two sections [to Student 10]. What shape are they?

Student 10: Square, two square sections…

Teacher: Hmm, 2 squares… What else [points to Student 11]?

Student 11: They’re rectangular.

Student 17: The dragon on the back is asymmetrical.

Teacher: What? [Exclaims and then turns and writes on the board.]

Student 18: No, it’s not

Teacher: The dragon on the back is asymmetrical… Can you tell us what asymmetrical means? (Figure 2)

Figure 2.

Polyomino.

The teacher’s goal was to lead the conversation as students came up with terms that describe dominoes. Their description stated that dominoes have texture, black and white spots, two line-separated sections, two squares, and eight corners.

Teacher: Two squares, okay. What we’re going to be working with are called… All the “ominos.”

Students: Ominos. All the dominoes…

Teacher: Did you know that there’s a bunch of ominos?

Student 33: Yes.

Student 34: Yeah.

Student 35: No.

Teacher: Well when you start what we did now [pointing to the board]. Was that dom… The dominoes… Do you think there are other ones?

Student 36: Yeah.

Teacher: Okay…

Student 37: Yes…

Teacher: So if I wrote…

Student 38: Pentomino.

Student 39: A what?

Teacher: If you saw a monomino, what in the world, I mean, what do you think about that one? For this one [circles two squares on the board], we wanted to make sure we saw that they were two squares. What do you think the monominoes would look like?

Student 40: One square.

Teacher: Ooh, it would look like one square, right?

Then the teacher drew the shape of one square above where she had written “monominoes” and drew two squares joined by a single side above where she had written “dominoes” for students to make the connection between monominoes and dominoes.

Teacher: You know about the pentominoes; how many squares?

Student 40: Five.

Teacher: Okay…

Teacher: Triominoes, how many do you think?

Student 50: Three.

Teacher: Okay, anybody thinks they have another one?

Student 43: A hexomino?

Teacher: Ooh! How many do you think are on the hexominoes?

Student 45: Eight.

Teacher: Hexominoes.

Student 46: Six!

Teacher: Okay, anybody thinks they have another one?

Student 50: Did you say there was a quadromino?

Teacher: Quadr…

Student 50: Four [holding up four fingers].

Teacher: I don’t know because my thing doesn’t have one… Yes [to Student 51]?

Student 50 was trying to associate quadr- with the number four in that instance; the teacher turned it into a learning moment by asking Student 50 about a game played on the computer with four blocks. The teacher turned to the board and wrote, “tetrominoes.” The teacher’s purpose was to reference the game Tetris, which the students knew and even played.

Teacher: Tetrominoes has how many?

Students 54: Four; Tetris!

The teacher then asked the students if there was any other way to make dominoes using two squares since they had made dominoes with two squares. Some students thought it was possible by only using one square, but they were making monominoes in the real sense, whereas others vertically placed a domino next to the original horizontal one.

Teacher: Ooh, she said to put it like this [horizontal and vertical placement].

Student 64: Right.

Teacher: Hold on a second, let’s think about math stuff. Is that the same [pointing to the two placements of a domino]?

Student 65: Yes.

Student 67: It’s all turned…

Teacher: Yes, it’s turned… Or rotation, remember those words?

3.3 Triominoes

The teacher used the learning moment to capture mathematics concepts like flip and turn that change the orientation without changing the figure. That is to say, one could move it anywhere, but it would still be the same. Using only two squares to make dominoes, students were asked to pick three squares to make triominoes. The teacher reminded students that a triomino consists of three equal-sized squares connected edge-to-edge.

Many students had a bit of an issue with making trinominoes when joining edges with the three squares. The task was to find all triominoes formed by the three squares. Most students came up with Figure 3, which sparked a conversation among classmates.

Figure 3.

Trinominoes first stage.

Teacher: Uh oh, remember when we talked about the dominoes [pointing to the board]? Wouldn’t that be the same if you put it up like this [gesturing towards a vertical arrangement of the figure]? Does it not look different?

Student 82: Yes

Teacher: Okay, no matter how I turn it or move or reflect, it is still the exact figure.

Most students moved one of the squares in the first picture and then continually moved one of the squares to generate the remaining four photos after creating the first two images in Figure 3. Some pupils, it seems, were unaware that they were producing identical triominoes and only shifting them around (rotating or reflecting the item). This sparked a debate on whether rotating, translating, or reflecting an entity changes its shape (length). As seen in Figure 4, some pupils understood that there are only two triominoes. This activity effectively brought the concept of isometry to the forefront (Figure 5).

Figure 4.

Triominoes final.

Figure 5.

Student responses.

3.4 Tetrominoes

Teacher: Okay, we just did the triominoes right now, I want you to grab one more piece, the same color. Okay, if we have four, what are those called?

Student 87: Tetrominoes.

Teacher: Tetrominoes, okay, build it and see… Raise your hand if you can tell me a way to make it.

Student 91: Uh, like, uh, up and down.

Teacher: Up [gesturing with her finger in a vertical motion]. Like a real long rectangle?

Student 89: You could have two at the bottom and two at the top.

Teacher: Two at the bottom [draws two squares] and two at the top [draws two more squares]. Looks like a big square; is that kind of what it’s looking like? Awesome. What else [points to Student 90]?

Student 90: L-shaped.

The students utilized their newly acquired visualization skills to link the four squares edge-to-edge to generate various combinations that comprise the set of tetrominoes. At that moment, examples in Figure 6 demonstrated what edge-to-edge implies for constructing tetrominoes or any other polyomino.

Figure 6.

Tetrominoes.

Student 90 began with a large square, then shifted one of the squares to make an L shape. The student then moved the top square to the left to form Z-shaped tetrominoes. Finally, one of the Z-shaped tetrominoes had one of its squares moved to become T-shaped tetrominoes.

Teacher: Put it on the table so I can see. I need to use my visualization skills [pointing to her eyes as she walks over to Student 90].

Student 90: [Shows the teacher his table.]

Teacher: Alright, fantastic!

The student argument continues as the lecturer sketches what the students come up with on the board. Student 90 anticipated that any more movement of the squares would lead to isometry. The students could tell it was the same picture even when the orientation reversed, and there were five alternative ways to construct the tetromino pieces.

3.5 Pentominoes

After the discussion and the exposure of students to dominoes, triominoes, and tetrominoes, the teacher wanted to take students to another level using pentominoes.

Teacher: As discussed in other polyominoes, a pentomino is a polygon made up of five equal-sized squares joined edge-to-edge. When rotations and reflections are not seen to be separate forms. Pentominoes… So right now, make sure you have five square pieces. I’m going to put a pentomino piece down on your table. Cool enough? Then, we’re going to try to find all of them. Students worked in groups of five at five separate tables to create pentominoes using the information they had been provided. Working in groups enabled students to contribute to the making of pentominoes from five squares and facilitated assistance for those who were having problems learning. The instructor exchanged the produced pentomino piece with a genuine pentomino while the kids worked out the parts. This continued until the kids found all twelve pentominoes. The instructor noted that the twelve pentominoes pieces represented alphabet letters and utilized FLIPNTUVWXYZ, which is the mnemonic for the twelve pentominoes. As opposed to other polyominoes, pentominoes feature special properties that may be employed in spatial thinking, particularly in puzzle games. Before the game, students were given twelve pentomino pieces and invited to play with them to build a rectangle. Using their eyes only, they were asked to try using their free hand to build the rectangle on the table using all the pieces. This puzzle was very intense and the students were deeply engrossed in what they were doing. During the making of the rectangles, the discussion at one table was particularly interesting.

Teacher: Can you talk more about the strategy used in making the rectangle?

Student 90: I know that since each piece has five squares and sixty squares in total, I had to decide which type of rectangle I could make. I decided to come up with a six by ten rectangle.

Teacher: Why six by ten?

Student 90: I just decided to because to me it seemed easy to work with.

Student 90 did have a strategy on how to make the rectangle by starting with the shortest side on the right and working to the left. The student made sure that the number of squares along the longest side added up to ten. The student then had to select which remaining pieces to use. That strategy paid off for the students who completed the square shown in Figure 7. The student also noted that at each point they had to keep asking themselves how many squares were needed at that particular point and also what impact this could have on the assembly of the rectangle going forward.

Figure 7.

Six by ten rectangle.

At the same table, four students came up with six-by-ten rectangles, but the pieces were located in different places. Students were curious, and even asked how many different arrangements of the pieces had been created for the six-by-ten rectangles.

But at other tables across the classroom, other groups of students were dealing with different dimensions of rectangles, which were five by twelve, four by fifteen, and three by twenty. It was noted that some students had different rectangle patterns but identical dimensions, whereas others had different dimensions but identical patterns. This elicited discussion in the classroom about how many possibilities might exist for each dimension.

Student 89: This seems familiar with what we learned last week about positive divisors of 60, right? The number 60 has 12 positive divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Student 89 was looking at possible rectangles that can be created by factors of 60 (1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10).

Teacher: Can we generate each one of those rectangles using 12 pieces of pentominoes?

Students: Yes, no, yes, yes, no.

Some students believed that all conceivable rectangles (1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10) could be built with the 12 pieces of pentominoes, while others disagreed. Some students pointed out that it is not feasible to make a 1 × 60 rectangle, which can only be done if all of the parts are formed like the letter “I.” However, a rectangle with dimensions of 2 × 30 cannot be constructed since parts like x are generated by three squares attached together, and hence cannot fit into the 2 × 30 rectangle size. One of the problems asks the participant to create a rectangle from twelve pentominoes; this is a frequent introduction to polyominoes. Because 60 has 12 divisors, there are six alternatives (1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10). Once again, due to the construction of certain pentominoes pieces, building a 1 × 60 rectangle or a 2 × 30 rectangle with pentominoes is unfeasible. Each of the other dimensions offers multiple options. Although these puzzles are rather basic, they are nonetheless highly effective for their essential didactic purpose.

3.6 Extension

Another set of simple-sounding yet challenging rectangle problems involves building two rectangles at the same time with twelve pentominoes. The better puzzles involving pentominoes are the ones that allow for multiple solutions for each rectangle. However, the best puzzles in this regard are those that don’t require the use of a rectangle at all. Here below is an example of an extension for this activity:

Form a 3 × 5 rectangle and a 5 × 9 rectangle with twelve pentominoes at the same time.

Form a 4 × 5 rectangle and a 4 × 10 rectangle with twelve pentominoes at the same time.

Form a 5 × 5 rectangle and a 5 × 7 rectangle with twelve pentominoes at the same time.

Form two 5 × 6 rectangles with twelve pentominoes at the same time.

4. Conclusions

The study of spatial visualization is significant because of the correlational and logical-intuitive evidence for its relevance to most technical and scientific jobs, including mathematics, science, art, and engineering. Involving students in activities that stimulated spatial skills are critical in equipping students with this needed skill. Activities can range from visual discrimination, games, and activities involving polyominoes. It allows students to create polyominoes pieces using squares manipulatives and then use the real polyominoes pieces for class activities createing a spirit of collaboration and promoting active learning.