When teaching mathematics, the most basic concepts are often the most challenging. For example, I had students in my special education class working on counting and creating sets of objects for numbers within ten for multiple years. At times, it felt like students were practicing the same counting activities repeatedly but weren’t making any progress. Studying the principles of Number Core was life-changing for me.
The Number Core is a set of conceptual understandings and knowledge students need to count successfully. It has four main parts: the number word list, written number symbols, cardinality, and one-to-one correspondence. It is important to remember that these are not learned in any specific order.
Let’s define these parts separately and then consider how they work together as students learn basic counting.
Number Word List
The number word list is simply the list of things we say in the rote counting sequence, “one, two, three, four…” Just as young children learn to sing the alphabet song without knowing the letters they are singing, the same happens with the number word list. The number word list always makes me think of my children as toddlers. When I wanted them to stop doing something, my last resort was to count to three. As soon as I started with “One,” my daughter would chime in and say, “Two, three.” She had no idea she was counting. She just knew this was the list of things Mommy said before she got mad.
It is important to gather data on how much of the number word list students know. Many students gradually grow the list they know over time. Commonly, there are intermittent gaps where the word list gets mixed up. This may happen at the last point of the student’s know number word list. Other times, a student may make an error and then be able to get back to the correct counting sequence.
Written Number Symbols
This segment of the Number Core is quite involved. The big idea is that the students know the symbol for the word “two” is “2.” This gets complicated though, because students also need to know when they count a set of two objects, that means “2” too. Understanding the written number symbols also means students can write a “2” or a close approximation when asked.
On a side note, I say “an approximation of the symbol” because many students struggle with number reversals. While we want to make a note and keep an eye on this, it is not something we are going to worry about until Third Grade. Until that point, it is still important to support students as they practice and develop the ability to consistently write the number symbols correctly.
Cardinality
Students have cardinality when they understand that the number they say when they point at the last object in a set tells them how many objects are in that set. Picture a student in your head sitting at a table with six cubes in front of them. The student touches each cube as they say, “One, two, three, four, five, six.” When the student is asked, “How many cubes?” they look at the cubes on the table for a moment and then say, “Three.” This student does not have cardinality. A different student does the same task, and when asked, “How many cubes?” this student responds by pointing to the last cube counted and says, “This is six.” This student also does not have cardinality since they think only the last cube is six and not that six is the whole set of cubes they counted. So cardinality is specifically when students understand that the last number they say when they point to the last object in the set tells how many are in the whole set.
One-to-one Correspondence
One-to-one Correspondence is when a student knows to say one number from the number word list for each object as they count. So, if a student counts that set of six cubes but touches the same cube twice, says multiple numbers for one object, or maybe doesn’t track the set as they count at all and say a bunch of numbers, they do not have one-to-one correspondence.
In the primary grades, one-to-one correspondence is being worked on in many different contexts. Students are learning to write one letter for each sound they hear in a word and say one word for each word they see when they read. So, one-to-one correspondence in counting is one way to see if students grasp this concept. Students with advanced concepts of number will demonstrate unitizing by creating equal groups of objects to count more efficiently. The first units students use when counting are often twos, fives, or tens.
Putting it All Together
Now, let’s consider a few student scenarios and see how the parts of the Number Core can help teachers determine the next steps in instruction.
Student 1: This student is sitting with six cubes on the table in front of them. As they counted the set, the students touched and moved each cube, saying one number. The count sounds like this, “One, two, three, five, six, eight.” When asked, “How many cubes?” Student 1 says, “Eight!”
Notes: This student demonstrated one-to-one correspondence as they touched and counted one number for each cube in the set. They also understood cardinality since they responded that the set has eight cubes (the number said when touching and counting the last object in the set). The number word list is part of the Number Core Student 1 struggles with.
Student 2: This student is shown a number card with the numeral six and asked, “Show me this (pointing to the number card) many cubes.” The student counts and sets out three cubes, saying one number for each cube as they set it out. When asked, “How many cubes?” Student 2 responds, “Three.”
Notes: This student demonstrated one-to-one correspondence as they counted and set out three cubes. They also showed they knew the number word list for at least one through three. The next step for this student would be to work on recognizing the written number symbols and continue expanding their conceptual understanding of numbers within ten.
Summary
When students count and create sets of objects, several understandings are integrated into the task. Students must say the correct rote counting sequence and the number word list. They must also say one number for each object as they count, demonstrating one-to-one correspondence. Students also need to understand that the number they say when they count the last object tells them how many are in the set, which is cardinality. Finally, students need to know the written number symbols to create a set for a given number or label a set they just counted with the correct total. Understanding these four parts of the Number means that teaching students to count can be a precise, data-driven process. Simply counting and making sets over and over will not make students better counters. Building a specific understanding of the Number Core will allow students to work on specific concepts that need instruction and practice so they can move forward.