# Cognitive Conflict in Technological Environment: Cognitive Process and Emotions through Intuitive Errors in Area, Perimeter and Volume

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Intuitive Errors

#### 2.2. Cognitive Conflict

#### 2.3. Emotions among Learning

#### 2.4. Technology in Teaching Mathematics

## 3. Research Goal and Questions

- What are the characteristics of the cognitive processes that appear when students interact with activities in a technological environment related to intuitive errors?
- What are the students’ emotions’ characteristics during the emergence and overcoming of the intuitive errors in a technological environment?

## 4. Method

#### 4.1. Participants

#### 4.2. Procedure

#### 4.3. The Chocolate Molds Activity

#### 4.3.1. Activity Context

#### 4.3.2. Task 1

#### 4.3.3. Task 2

#### 4.3.4. Task 3

#### 4.4. Data Sources

#### 4.5. Data Analyses

## 5. Finding

#### 5.1. Cognitive and Emotion Processes during the Intuitive Error in the Three Tasks

- (5) Adam: I think the amount of the chocolate will be the same in the two cylinders; We made them by the same rectangle (looks to the other students in the group)
- (7) Ahmad: it will be the same amount (with his eyes on the computer screen, and shakes his head); in one case it will be thin or wide, but the same amount.
- (21) Ahmad: Yeah, look how…, the cylinders (while the applet builds the cylinders)

- (105) Adam: the same rope is the same circumference; the cups cover the shape we Produce (in a confident tone)
- (107) Ahmad: the number of the cups is the area (looking to other students with comfortable and confident tone)
- (109) Ahmad: Sure, it is equal, the rope has not changed. I am sure
- (111) Adel: We can frame different shape, but it still the same rope; yes I think so (shakes his head)
- (112) Ahmad: it is the same circumference in all shapes, maybe sometimes we need to divide the cups, it will not be a whole number
- (113) Adam: it did not matter, you will have the complementing parts

- (206) Mohammed: how we know, we do not have the dimensions of the cup or the box
- (207) Rami: it does not matter; we know that the cup has equal lengths
- (208) Sameer: we do the same process increased four sides and increased four sides by quarter.
- (209) Rami: I think the seam extended and curtailed in the same mount
- (210) Mohammed: I think we will have errors like the other activities we did (smiles)
- (211) Rami: I think no; this time, they are equal. (shakes his head with confident tone)

#### 5.2. Cognitive Processes and Emotions during Revealing the Intuitive Errors in the Three Tasks

- (83) Ahmad: Ha; the size is not the same thing? (with a surprised tone) (see Figure 2)
- (84) Adam: what? 102? 56? (he means the volume of the cylinders; he talks with a surprised tone)
- (85) Adel: However, how? why are they different? (he means the cylinders’ volume, he looks at the other students and talks in a low and sad tone)
- (86) Ahmad: How this happen, it the exact dimensions (he points to the computer screen)
- (87) Adel: The same rectangle. Each time we rounded differently; why the volume different.

- (155) Ahamd: It is different! (with a surprised tone, deeply looking at the screen computer and then to other students) (see Figure 3)
- (156) Adam: oh, all of them! (with a surprised tone), they (the different rectangle) not equal
- (157) Ahmed: Haa, it means we are wrong (knocks with his hand on the table).
- (158) Adel: uh, may we try a different length of the rope? (with a non-confident tone)
- (159) Ahmed: Ok; I changed (changed the rope length)
- (160) Adel: uh, it does not matter; the area is not equal (with a surprised tone)
- (161) Adam: It is different; all of them (rectangles).

- (210) Adam: it is impossible to be wrong (deeply looking at the computer screen (see Figure 4)
- (211) Adel: it is different in all of them (with a surprised tone)
- (213) Ahmad: I do not understand (with an uncomfortable tone)

#### 5.3. Cognitive Processes and Emotions during Overcoming Intuitive Errors in the Three Tasks

- (101) Adel: always the volumes are different (indicates to computer screen and then looks to the other students)
- (103) Adam: the thin one is always smaller (the volume of the thin cylinder is smaller than the wide one)
- (104) Adel: the radius here (the thin cylinder) is smaller than this (the wide cylinder)
- (105) Ahmad: Ahh, the area here is bigger (the bases of the wide cylinder, with loud tone)
- (106) Adam: but here (the thin cylinder), the highest is bigger
- (107) Ahmad: but the radius we squired! (in confident tone talking to Adam)
- (109) Adam: Ah, this will affect more in the volume (with a smile)

- (170) Muhammad: Compare this rectangle and this rectangle. Sure, that they will not have the same area.
- (171) Rami: why you are sure? (looks to Muhammed)
- (172) Sameer: It is so thin (indicating one of the rectangles) it will include a small Number of cups, but here (another rectangle), it will include more
- (174) Muhammed: Yes, it is clear (shakes his head)
- (176) Rami: look, it is the biggest area (with a confident tone)
- (177) Muhammed: it is square; try another length (request from Rami)
- (178) Rami: you see (produced different rectangles) the biggest is the square
- (181) Sameer: always the square is the biggest area for the same circumference
- (182) Rami: I think it is logic (shakes his head); if we closed the sides, there would be a small area. The square is the shape that we keep each of the parallel sides with the same distance.
- (183) Muhammed: Yes, so it includes more units (with a confident tone)
- (184) Rami: it is means the same area.

- (262) Lean: how much it smaller, it is different each time (serious tone)
- (263) Noor: the difference is to do subtraction between them
- (265) Maise: in all the cases, it is the same number, look here (the difference between the volume), Ah, it always the same (opens her mouth)
- (266) Noor: yes, it does not matter the dimensions. It always the same result (shakes his head)
- (268) Maise: I think it because always the multiplication of these sides (the one which was extended, and the one which was reduced)
- (269) Noor: Ah, look this (the side that is constant) and always the multiplication of these sides are 15\16 (loud tone)
- (270) Maise: it misses 1\16, so it always smaller 1\16
- (272) Lean: yes, it does not matter what the sides are
- (273) Maise: but it is a quarter?
- (274) Noor: may we can try fifth (short pause) it 1\25 smaller
- (275) Maise: May try sixth? (with an excited tone)
- (276) Noor: it is 1\36
- (277) Maise: I think seventh is 1\49 (excited tone and smiling)
- (278) Lean: OHH, yes. We found the method! (knocks on the table with smiling)

## 6. Discussion

## 7. Implication

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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Category | Examples of Sub-Categories | Examples from Participant Discussions and Behaviors |
---|---|---|

Positive emotions | Confidence | Shaking head as accepting |

Enjoyment | Specific words such as “OHH”, “YEAH” | |

Comfort | Relaxed tone of voice Smiling | |

Negative emotions | Confusion | Putting a hand on the head or the face |

Non-comfortable | Loud voice or sad tone of voice |

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**MDPI and ACS Style**

Shahbari, J.A.
Cognitive Conflict in Technological Environment: Cognitive Process and Emotions through Intuitive Errors in Area, Perimeter and Volume. *Mathematics* **2021**, *9*, 1672.
https://doi.org/10.3390/math9141672

**AMA Style**

Shahbari JA.
Cognitive Conflict in Technological Environment: Cognitive Process and Emotions through Intuitive Errors in Area, Perimeter and Volume. *Mathematics*. 2021; 9(14):1672.
https://doi.org/10.3390/math9141672

**Chicago/Turabian Style**

Shahbari, Juhaina Awawdeh.
2021. "Cognitive Conflict in Technological Environment: Cognitive Process and Emotions through Intuitive Errors in Area, Perimeter and Volume" *Mathematics* 9, no. 14: 1672.
https://doi.org/10.3390/math9141672