As soon as children enter elementary school, they start challenging others to do calculations they consider complex. “What’s 1000 + 1000?” they ask each other, proud to show off their math skills.
Although mental arithmetic has been taught since the dawn of time, teaching methods have changed significantly. Over the past few decades, teaching methods have evolved from making students memorize number facts to teaching them strategies that research shows are much more sustainable over time.
“Do not subject any student to fact drills unless the student has developed efficient strategies for the facts being practiced… Short-term gains are almost certain to be lost over time. Practice prior to development of efficient methods is simply a waste of precious instruction time.” (Van de Walle, 2001, p. 144)
What should teachers do?
Students need to develop number sense before they can start doing calculations. Through mathematical exploration, manipulation, and discussions in class, they can develop their own strategies for doing arithmetic, first in context (problem solving), then in non-contextualized exercises.
Number sense
The National Council of Teachers of Mathematics establishes a link between the conceptual understanding of numbers and academic success in math (2020). Students with a good sense of numbers and operations are better able to tackle challenges and determine if their answers make sense.
In practice, number sense is developed by visually representing numbers with different mediums (chips, blocks, fingers, number lines, number grids). This way, children can understand that each number corresponds to an object that they count and can establish links between numbers (ascending order, descending order, smaller, bigger, equal, more, less, etc.).
Practicing number manipulation also allows children to use the numbers 5 and 10 as anchors for many physical and mental manipulations.
Recall strategies
Once children have acquired a good number sense through many manipulation exercises and engaging discussions in class, they can start learning number facts and recall strategies.
As for number sense, students should start by using objects, before gradually getting used to mental math.
The number facts +1, -1, +2, and -2 are a good place to start. This makes students aware of numerical order and how it affects operations. It’s also a good time to introduce students to the commutative property, or the idea that the order of the numbers in addition and multiplication doesn’t matter: 3 + 1 = 1 + 3.
The property can also be illustrated with number houses. Number houses can be used to write down all addition operations that have the same sum. The house can be divided in the middle, highlighting the commutativity of addition with the two columns.
The doubles strategy is a good second step that will help students learn other strategies. There are very useful illustrations that allow students to visualize doubles and can help more visual learners (e.g., an insect has 3 legs on one side and 3 legs on the other side, so it has 6 legs).
Once they’ve mastered doubles, children can more easily understand the neighboring number strategy: to find the answer to 5 + 6, students need to do the equation 5 + 5 in their heads and add 1.
As previously mentioned, the numbers 5 and 10 are good anchors for doing calculations. Using boxes containing 10 chips each, students can try to group the tens together to calculate more easily. For example, for 7 + 4, they can add 4 chips to a box of 10 that had 7 chips in it and see that the total is 11 (1 full box of 10 + 1 chip).
Mastering number facts
“Timed tests make no instructional sense. Children who perform well under time pressure display their skills. Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure. In addition, children can become fearful and negative toward their math learning.” (Burns, 2000, p. 157)
Remember that efficient strategies are needed to master number facts in the long term. Teachers can ensure their students are learning these strategies properly through discussion, observation, and production.
