Most people believe that counting is the foundational skill to understand math. So, beginning at a young age, children are encouraged to memorize a long string of counting words. Then, children learn to use those words to count objects. We give them a group of items and teach them to touch each item while saying the next counting word. Once the child has completed this ritual, they respond to the question, “How many items are there?” by restating the last counting word they spoke.
For example, when we ask the young student to add 6 and 5, we have them start by counting out 6 blocks. Then, the student must count out 5 more blocks. Finally, they are instructed to count how many blocks they have. Most of the time, the child starts by counting all the blocks from 1 rather than just starting at 6. Why can’t the child just count on? Because the young child finds it difficult to start counting at 6, they start back at 1. As adults, we may think counting on is an easy skill. However, it isn’t. Consider the nursery rhyme “Jack and Jill.” What word comes after “hill”? Did you know, or did you have to start at the beginning to figure it out?
To further help you understand how a child experiences number sequences, consider using letters instead of numbers. For this activity, let’s replace the number 1 with the letter A, 2 with B, 3 with C, and so forth. Then, using the counting model, add F + E by counting out F blocks (A, B, C, D, E, F). Then, count out E more blocks (A, B, C, D, E). How many blocks are there? You can either count all the blocks or count on from block F to find that the answer is K.
What if you didn’t have counters? How would you count on from F? You could use your fingers to add E more to F by raising one finger for A while saying G to keep track of your tally. Then you can raise another finger for B while saying H. You will continue this process until you get to E fingers and find that the final answer is K. What if you could not use your fingers? You would be forced to keep track mentally: A of E is G, B of E is H, C of E is I, D of E is J, and E of E is K. Wow! That is tedious and confusing!
Now that you know how to add, you can easily memorize the facts, right? Answer these as quickly as you can. What is D + C? Or H + G? How about F + C? See how difficult this is? Just because a student knows the process does not mean they can quickly get to the solution.
As a result, we decide to provide more practice opportunities by using flashcards. When using flashcards, students need frequent and regular practice, or they will quickly forget what they have learned. Unfortunately, flashcards are not an effective practice model for one out of seven children. They especially are ineffective for students with learning challenges. In fact, the only people who enjoy flashcards are those who do not need them because they already know the math facts. Unfortunately, the use of flashcards is one reason why many students develop a dislike for math and fail to thrive in their math classes.
There is a better way: Subitizing. Subitizing is the ability to recognize a quantity without counting. Studies have shown that infants as young as five months can recognize up to three objects. Children as young as three can subitize up to five objects once they understand that five includes a middle. Unlike counting, subitizing enables the student to see the whole and the individual items at the same time.
When subitizing 6 through 10, quantities can be easily recognized when separated into two groups: 5 and the rest. In other words, you can easily recognize the quantity of 8 when split into two groups: 5 and 3 more. For thousands of years, people have been grouping in fives, most likely because our hands have five fingers. Consider the Roman numerals, tally marks, the musical staff, and even the Chinese abacus. They all include groupings of five.
One significant math skill is the ability to visualize or see something in the mind. Without grouping, can you visualize eight identical apples? For most people, this is an impossible task. However, can you visualize five red apples and three green ones? Most people can visualize larger quantities when they are grouped into fives. There is a strong connection between subitizing and visualizing. If something can be subitized, it can also be visualized.
Try this. Looking at the left image below this paragraph, can you tell immediately how many yellow tiles there are? Do you know for sure? Now, look at the tiles on the left? Can you see immediately how many tiles there are? Most people can see right away that there are eight tiles because of the color change.
Subitizing and visualizing are related skills. You can visualize what can be subitized.
In some countries, children are not allowed to count when solving addition problems. Instead, they learn to modify the problems mentally. For example, when adding 4 + 3, students are taught to imagine a group of 4 blocks and another group of 3 blocks. Then, they imagine taking one away from the group of 3 blocks and giving it to the group of 4. This process modifies the addition problem to one that the student can easily solve: 5 + 2, which is 7.
Recent studies have suggested that a young child’s understanding of number sense predicts their future ability to excel in math. As instructors, we should replace counting with subtizing, giving our students a right start in mathematics.
