In the overview module, we discuss the connections between executive functioning (EF) skills and early math learning in general (see Overview of Executive Functioning and Math). Here we discuss the role of executive functioning during mathematical operations. When learning the foundational skills that underlie addition and subtraction, children learn that sets of objects can be manipulated to solve problems (e.g. a set of five bears can be added to a set of two bears to make a set of seven bears), setting the stage for more complex math learning. As mentioned in The Mathematics of Operations, mathematical operations involve an understanding of patterning and structure, decomposition, place-value and equivalence, all of which call upon young children’s executive functioning skills.

• Patterning and Structure: Many operational patterns and structures are dictated by a set of number properties. For example, the commutative property of addition dictates that the sum of a + b must be equal to the sum of b + a. When young children are first learning how to combine sets, they typically learn that 3 objects plus 1 object equals 4 objects. In order to understand that 3 objects added to 1 object also equals 4 objects, they must use their cognitive flexibility skills to realize that whether a set of one is added to a set of 3 objects or a set of 3 objects is added to a set of one object, the total of both sets together equals 4.
   
• Decomposition: Decomposition is a common strategy for solving arithmetic problems that involve the transformation of one problem into a series of smaller problems. For example, when creating a set of thirteen objects, children may create a set of ten and then add on a set of three. Importantly, when decomposing arithmetic problems, children must use their working memory skills to hold multiple quantities in mind at once. For example, if children are asked to find different sets of numbers that sum to 10, they have to manipulate different combinations of numbers (e.g., 2 and 8, 3 and 7, 1 and 9). However, before children can successfully use decomposition strategies, they must first understand that different numbers can be added together to arrive at the same sum. Understanding, this basic aspect of decomposition (that the combination of different numbers can result in the same sum) involves cognitive flexibility.
   
• Place-Value: In the English number word sequence, the pattern is not regular until the second decade (20). In particular, the numbers one, two, three, and five do not directly map onto the number words of eleven, twelve, thirteen, and fifteen. Children need to use their cognitive flexibility skills to understand that these “teen” words need special treatment and inhibit the urge to make them regular (which can result in interesting combinations such twelveteen!).
   
• Equivalence: The concept of equivalence in operations means that the equation a + b = c is equivalent to the equation c = a + b. In building the foundations to these skills, children come to understand that a set of 2 objects combined with a set of 3 objects makes a set of 5 objects and also come to understand that a set of 5 objects can be separated into sets of 2 and 3 objects. Later this knowledge can support the use of cognitive flexibility skills in translating this to equations such as 2 + 3 = 5 and 5 = 2 + 3, which supports deeper understanding of equivalence.   

The examples above suggest that executive function abilities are important for the conceptual understanding of mathematical operations, and that doing math problems utilizes (or employs) executive functions. As such, early childhood educators should consider activities that involve mathematical operations as opportunities not only to support the development of important mathematical skills but executive functioning skills as well. 

Below we offer some tips for how early childhood educators can incorporate more explicit EF support into activities involving mathematical operations.  

Working Memory: Students practice working memory skills when they are required to hold and/or manipulate multiple pieces of information in mind over short periods of time in order to solve problems. In fact, many arithmetic activities offer opportunities for children to develop their working memory. For example:

• The teacher may start out with a fair-sharing activity by saying “If you have 6 crackers and you share them with Anna so you each have the same number, how many will you have and how many will Anna have?”
• Teachers can help children solve problems during play by asking questions such as, “How many more blocks to you need to make that blue tower equal to the height of your purple tower?” Children need to hold the number of blocks in the purple tower in mind as they figure out how many more blue blocks are needed.

Inhibitory Control: Students practice inhibitory control skills when they engage in activities that require them to wait their turn and/or think before they act.

• Teachers can support inhibitory control by engaging their students in rule switching. For example, the teacher may ask the student to roll two dice and then ask “What is the total of the two dice?” After repeating this a few times, the teacher may then ask “How many would you get if you subtracted the smaller die from the bigger die?” In this way, the child has to think before s/he acts in order to inhibit their original response (to add the number of pips on both dice) and to follow the new rule (to subtract). Note that this activity also supports cognitive flexibility as the child now has to approach the same activity in a different way.

Cognitive Flexibility: Teachers can support cognitive flexibility by encouraging students to make comparisons, shift perspectives and to approach activities in different ways. For example:

• Teachers can support cognitive flexibility in their students by engaging them in games that involve both addition and subtraction. For example, the teacher could organize a game in which the students have to show with objects the result of two plus two more, minus one, plus three, and so on. Asking children to state the resulting quantity at each step reinforces addition and subtraction understandings as well as additional executive functioning skills such as inhibitory control and working memory.

Planning and Organization: Students practice planning and organization skills when they engage in multi-part activities that require the organization of materials and/or adherence to a sequence of steps.

• Teachers can support planning and organization skills by encouraging students to think through the steps of performing mathematical operations. For example, the teacher may ask a child to explain how they figured out “how many altogether” when adding two sets of objects together.