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Addition and subtraction: additive operations

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Addition and subtraction are two of the basic operations that students starting working on very early in kindergarten and then in more depth throughout the first grades of elementary school. We will move past the exact calculation algorithm and focus on explaining what addition and subtraction means and what it means to learn how to do these operations.

To do so, we will go over one of the teacher resources that we have in the Manager, the training capsules, in which Cecilia Calvo and other members of the teaching team dissect the processes involved in learning addition and subtraction.

We should not confuse carrying out an algorithm skillfully with knowing how to add or subtract. Mathematics is not a list of procedures or formulas that students should learn, it is a way of thinking, of making thinking more efficient. This is a vision of mathematics that is closer to how science is viewed by society, and it is the vision that we, at Innovamat, want to transmit and that forms the basis of these capsules.

The training capsules, dedicated to basic arithmetic operations, are an archive of reflections and arguments that are supported by studies and specialized literature and that favor meaningful learning of operations and, they can also be used as a small resource bank.

Here are 3 of the training capsules:

And here is a summary of them:

The idea of counting and addition and subtraction strategies

In capsule 2 we explain how to construct the basic ideas of addition and subtraction based on counting. We consider counting (ascending and descending) as a transitional base for learning these two operations, so that children can later help themselves to count when doing deduction chains (what we call known facts – derived facts). We also propose various addition word problems, which aim to promote different solving strategies and encourage mathematical conversation in class to evaluate, in terms of efficiency, each of the strategies.

We also explain the learning process in detail: we first handle objects that we can count, for example, cubes; then, we move to a higher level of abstraction by limiting the use of manipulative material by offering students the possibility of leaving counting behind. And finally, we build more efficient strategies, with conversation as the backbone to encourage the sharing of knowledge.

There are different strategies to leave counting behind. For example, the known facts-derived facts strategy, in which we used already known additions and subtractions to work out others. For example: 8 + 9 is one more than 8 + 8 = 16, therefore 8 + 9 = 17. This strategy, like all others, is introduced using manipulatives: I build two bars of 8 cubes and ask myself how many cubes I have. Then I add one more cube to one of the bars. In this case, we don’t need to count them again, I know I have 16+1. And once I’ve used the manipulatives, I can represent this strategy on paper to internalize it. Then, if it’s internalized, I can use it repeatedly, in an abstract way, to calculate mentally.

Another strategy is making 10. This strategy is linked to jumping along the number line, a key model for developing mental math, and consists of using the number ten to jump more efficiently. To finish, we can also talk about the decomposition strategy, which helps us when working on the calculation written as a vertical algorithm.

A classroom example

Jumping along the number line in the 0-100 range

In Capsule 3 we build on additive operations in the 0-100 range using the knowledge acquired from different strategies:

  • Known facts-derived facts: For example, I know that 3 and 3 is 6, and adding one more is 7.
  • Addition Squares: Ideal in situations where we instantly see two separate groups of elements and the total amount of elements. For example, in the case of the image: a group of 5 cookies, a group of 2 cookies and the total of 7.

What is important to highlight about the squares is that they encapsulate many relationships between the 3 numbers that make them up. Not only do we find 5 + 2 = 7, but also 7 − 2 = 5 and 7 − 5 = 2. Working with the squares allows us to internalize and automate these important relationships when calculating additions and subtractions with digits.

In addition, we present the close relationship between addition and subtraction in order to maximize the bank of known facts and derived facts that allows us to substitute memory-based and mechanical learning for reasoning.

Subtraction with regrouping and how to build it in the classroom in three steps

Lastly, throughout capsule 7 we reflect on the algorithms that we traditionally teach students for subtraction with regrouping. We find that they are not unique and that there is a great cultural factor behind them, and we ask students to be flexible when using them.

We also offer resources to explore the transparency of algorithms further (i.e. the need of explaining why the procedure works when making calculations) and to provide examples of how to build them in the classroom in 3 steps:

  1. We work with manipulatives.
  2. We represent this material graphically (an important bridge between manipulation and abstraction).
  3. We approach abstraction: we turn it into symbolic language.

With these resources, we highlight that there is not only one way to do each of the elementary arithmetic operations, and we provide some guidelines on how to move towards building flexible and transparent algorithms. Because learning addition or subtraction is more than learning a standard algorithm, and there are many other strategies that can be used to master these operations (jumping along the number line, number decomposition, transparent algorithms…).

  • Laura Ansorena

    Trained in Architecture and is a teacher by vocation. She likes beautiful, real things, like math. She is currently a teacher of Mathematics, Technical Drawing and Design at Aula Escola Europea. She combines teaching with a collaboration in the Innovamat Teaching department.

  • Cecilia Calvo

    I’m Cecilia and I was born in Montevideo. I have a degree in Mathematics from the Sciences faculty from Universidad de la República in Uruguay, from the first year of my degree I knew that mathematic education was the path for me. In 1994, I wanted to continue my adventure in Barcelona where I started my postgraduate in Experimental Sciences and Mathematics Education in Universidad Autónoma de Barcelona. I completed my doctorate in 2001. The classroom has always been my second home. After 30 years working with students, I currently dedicate part of my work to initial and continuous mathematics teacher training and, since 2017, the development of materials for student’s use in Innovamat. I like to think that I am always learning.

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