# Developing Number Sense: An Approach to Initiate Algebraic Thinking in Primary Education

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## Abstract

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## 1. Introduction

#### 1.1. Theoretical Foundation

#### 1.1.1. Visualization, Manipulation and Number-Space Linking

#### 1.1.2. Development of Relational Thinking

#### 1.1.3. Development of Flexible Quantitative Thinking

#### 1.1.4. Understanding and Generalization of Patterns

#### 1.1.5. Non-Algorithmic Calculation

## 2. Materials and Methods

#### 2.1. Teaching Materials

- Number line/tape (Figure 1): It is a tape which facilitates the appropriation of numbers as a linearly ordered, continuous and expandable sequence. It starts with 0, number used as starting point, absence of accounting elements or total loss, and end with 100, as a gateway to the numbers with hundreds that will be studied later on.

- Numerical panel: it presents the numbers from zero to ninety-nine by families, enabling new possibilities of analysis and relation.

- Numbering box (Figure 4): it is a box with the separation of units, tens and hundreds. Additionally, numerous plastic sticks and red and green rubber bands are used. The whole resource is essential for understanding and working with our decimal number system. Through the use of the sticks and rubber bands, teachers will introduce students to the characteristics and rules of the system. They will be able to see equivalences and will manage to differentiate between different parts of the number without losing the idea of totality.

#### 2.2. Transversal Issues

## 3. Results

#### 3.1. Relate a Number with the Previous Ten and the Next Ten

Teacher: How can we relate this number to fifty? How can we go from one to the other with operations?

Student: If we give one to fifty, we reach fifty-one.

Student: And if we take one from fifty-one, we go back to fifty.

Teacher: And with the next ten, how can we relate it to the number fifty-one?

Student: If we add nine, we get to sixty.

Student: And if we take away nine, we return to fifty-one.

#### 3.2. Verify and Generalize Numerical and Arithmetic Patterns

Teacher: If I add ten to ninety, I’ll reach one hundred, and a hundred minus ninety returns to ten... how do we express this relationship between ten and one hundred?

#### 3.3. Know and Understand the Decimal Number System

Teacher: As we can see, fifty-two is five tens and two units, but we can also say that it is fifty units and two more units. How can we express this number with a sum?

Student: We can express it like this: 52 = 50 + 2.

Teacher: I suggest you do the operation 32 − 17 with the box.

#### 3.4. Skip Kangaroo!

Student: If it takes a jump of five, it reaches thirty-three

Teacher: How will we express this by writing what you have just said? (The teacher writes on the board at the same time they tell what kangaroo has made)

Teacher: From what number has the kangaroo always jumped? What jumps has it made? With what jumps did it stay close to the start? With which did it get very far?...

#### 3.5. The Frog Saltarina

Teacher: Today, Saltarina adds seven, where will it stop, and which number will it reach?

Student: It will jump five to land on eighty and later it will jump two.

Teacher: Well done! Let’s check it!

The teacher performs the two movements with Saltarina, stopping at eighty and then reaching eighty-two.

Teacher: It has reached eighty-two! Let’s write what has happened.

Students: Saltarina was on seventy-five (the teacher writes 75), first it has made a leap forward of five to reach eighty (she writes +5) and then it has made another leap of two (+2). It has reached eighty-two (Figure 29).

Teacher: It has reached eighty-two!... What if it now adds nine?

Student: Now it has to jump eight to ninety and then one (Figure 30).

Teacher: And how do we narrate it with a calculation?

## 4. Discussion

- The importance of number sense development in the first years of mathematical learning, and the importance in this process of achieving significant learning of the decimal numbering system and a comprehensive and relational management of arithmetic operations and their properties, as suggested by [9,44]. All this is aligned with previous research that highlights that small changes in traditional arithmetic practice and learning environment can be key to conceptual understanding [7,45].
- The importance of algebra in secondary education, as a gateway to symbolic thinking in Mathematics, and the difficulties traditionally faced by both teachers and students of this educative level, in order to develop the competencies expected in this area.
- The steps that have been taken in recent years in research in mathematics education in relation to early algebra, in favor of an appropriate transition from concrete arithmetic to the symbolic language of algebra and its positive consequences in favor of abstract mathematical reasoning. Furthermore, in this sense, the relative abundance of specific examples, but the lack and the need to have reference methodological models that, with greater depth, make possible a real breakthrough in the context of practice in the classroom [13,14].
- Finally, the conviction that in the first years of school learning it is necessary to use manipulative resources for the construction of mathematical knowledge in general and the development of number sense in particular [10].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 22.**Process to take away 17 from 32: firstly, the student removes 2 units (

**a**), secondly, one ten (

**b**) and finally 5 units (

**c**) by removing a rubber band of one ten.

**Figure 26.**Kangaroo child activity 1: “Write additions and subtractions in such a way that the kangaroo overcomes number 60”.

**Figure 27.**Kangaroo child activity 2: “Write additions and subtractions in such a way that the kangaroo overcomes number 60”.

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**MDPI and ACS Style**

Adamuz-Povedano, N.; Fernández-Ahumada, E.; García-Pérez, M.T.; Montejo-Gámez, J.
Developing Number Sense: An Approach to Initiate Algebraic Thinking in Primary Education. *Mathematics* **2021**, *9*, 518.
https://doi.org/10.3390/math9050518

**AMA Style**

Adamuz-Povedano N, Fernández-Ahumada E, García-Pérez MT, Montejo-Gámez J.
Developing Number Sense: An Approach to Initiate Algebraic Thinking in Primary Education. *Mathematics*. 2021; 9(5):518.
https://doi.org/10.3390/math9050518

**Chicago/Turabian Style**

Adamuz-Povedano, Natividad, Elvira Fernández-Ahumada, M. Teresa García-Pérez, and Jesús Montejo-Gámez.
2021. "Developing Number Sense: An Approach to Initiate Algebraic Thinking in Primary Education" *Mathematics* 9, no. 5: 518.
https://doi.org/10.3390/math9050518