I know church is supposed to be a reflective place. Can I help it if I happen to reflect on math? I hope not! As we were leaving one Sunday morning, there was a significant back up on the stairs. This is frequently caused by the very dainty, and somewhat fragile church goers who live in the retirement home next door. On this occasion however, it was a different kind of dainty parishioner – a toddler. As she carefully graced each step with both feet, her dad peered behind his shoulder and gave us all the, “I’m so sorry but I know you understand.” look. He then returned his attention to his daughter and began counting.
The act of counting steps, or anything tangible, is a vital component to supporting young children while they mathematize their world. Mathematizing is really just about bringing out the math that is inherent in the world and space around us. That things (anything really) can be counted is mathematizing. Kids mathematize when, given the option of a portion of cake, choose what they perceive as the larger portion. They learn about volume and dimensions when trying to build towers with varying sized sets of blocks. They mathematize when they’ve calculated that there is only one “fun” swing on the playground and the likelihood of loosing it is good so giving it up to play on something else is not an option!
Early counting and grouping is of particular importance. The action of that father counting steps with his toddler supports her development of cardinality. Cardinality is the idea that number and quantity are related. Each number represents a set of that many things. While this is obvious to you and me, it is not clear to the youngest mathematicians of our world. Watch a very little child, 2 or 3 years old, try to count a set of objects. He may understand the idea that he is supposed to say the count sequence while pointing to objects but he may not yet know that each number he says has to correspond to one of the items. And not only that, each number has to correspond to a different item. He doesn’t know you can’t count it twice! This one-to-one correspondence develops over time and through repeated opportunities to practice with guidance. As the toddler on the steps felt each count underfoot, she was developing one-to-one correspondence and cardinality.
These ideas of cardinality and one-to-one correspondence are components of the kindergarten Common Core State Standards for Mathematics. They are really quite basic but so vital for mathematical proficiency. While they “live” in the world of kindergarten, they are skills that can and should be developed much earlier. At home and at daycare, adults can help children mathematize their world by subtly applying the count sequence to objects. Imagine all the times you could say, “Let’s count them!” Rocks, legos, beads, toys, shoes, birds, swings, diapers, grapes, forks, blocks, friends, etc. Adults can easily teach children to touch and count each object and to help them distinguish between the counted and uncounted by demonstrating pushing the counted collection aside, one by one. While one-to-one correspondence takes time and fine motor skills to develop (so don’t fret if it takes awhile), modeling of this behavior is invaluable.
One more thought on cardinality. When children finish counting a set of objects and are done saying the count sequence, we assume that they understand that the last number said represents the total in the group. This is not necessarily the case. If you follow up a counting sequence by asking, “So how many?” you may notice that your child repeats the count sequence. This is a good indication that he does not yet understand this component of cardinality. You can’t force him understanding, that will take time and experience, but you can model it yourself. When you model you can say, “1, 2, 3, 4, 5. Oh, there are 5 markers!” This indicates to your child which of the numbers represents the total of the group.