Most people think of counting as the foundation of math. So we teach kids to memorize the long string of 100 counting words. Next, to count objects, we teach them to touch each block in turn while saying the next word in the string. After completing this counting ritual, the child must be ready to answer the “how many” question by repeating the last number spoken.

To add 6 and 5, we tell the child to count out 6 blocks, then 5 more blocks, and finally to count all of them. How many times have we been annoyed when the child finds the sum by counting starting at 1, instead of 6? To start at 6 means you have to know what comes after 6 without starting from the beginning. This isn’t so easy: In the nursery rhyme “Jack and Jill,” what word comes after “hill”? Did you know, or did you start from the beginning?

To experience adding as a child, let’s use the alphabet instead of numbers; A is 1, B is 2, C is 3, and so on. Now add F + E. First count out F counters (A, B, C, D, E, F). Next count out E counters (A, B, C, D, E). How much is it altogether? Either count all or count on from F to get the answer, which is K.

If you didn’t have counters, how would you count from F? You might use your fingers: raise one finger for A, while saying G. Then raise another finger for B, while saying H. Continue until you raise E fingers and say K. If fingers are forbidden, a tedious mental dialogue takes places: A is G, B is H, C is I, D is J, and E is K. Goodness! What a lot of work.

Now that you learned how to add, memorize the facts. Quick: What is C + D? What is H + G? What is F + C? And out come the flash cards, which by the way don’t work for one out of seven children — especially those with learning challenges. Even for the other six out of seven children, the memorized results are short-lived, needing frequent review. The only person who likes flash cards is the person who doesn’t need them. Sadly, flash cards often are the root cause for dislike of and failure to thrive in math.

Happily, there is another way: subitizing, which is the quick recognition of quantity without counting. Even 5-month-old infants can subitize up to three objects. Three-year-olds can subitize up to five objects, if they are taught that five has a middle. Subitizing, unlike counting, allows the child to simultaneously see the whole and the individual objects.

The simplest way to subitize quantities from 6 to 10 is to group them into a group of five and the rest. Grouping in fives goes back thousands of years, probably because of our hands. For example, Roman numerals, tally marks, Chinese abacus, and the musical staff are all grouped in fives.

An important part of math is visualizing, seeing something in your mind. Try to imagine a row of eight identical apples without any grouping — virtually impossible. Now imagine five red apples and three green apples. That most people can do. You could say our brains were created to match one hand, not two. There is a relationship between subitizing and visualizing. What that can be subitized can be visualized.

Children in some countries are discouraged from counting for addition. They are taught to work with quantities mentally. For example, to add 4 + 3, they’re taught to mentally imagine a group of 4 tiles and another group of 3 tiles. Next they mentally take 1 tile from the group of 3 and give it to the group of 4. This changes the problem into 5 + 2, which they know is 7.

Experts tell us that young children’s number sense is a good predictor of their later math ability. Let’s replace counting with subitizing and visualizing to give our kids a right start in math.

Brilliant. I wish I had this when I was a kid.

Thank you for such an insightful blog post! I so look forward to reading “B” next, followed by “C”, and then “D”… Your knowledge is truly a gift for many to glean from, Dr. Cotter.