Structuring School Opportunities

Arithmetic Skills

Children have four years — ages 3, 4, 5, & 6 — to construct a vision of five-ness, so we can cool our jets, as long as we understand what arithmetic skills and concepts they acquire.

Abilities in arithmetic can be classified as skills (practiced abilities), concepts (mental understandings that are constructed over time), and making connections to symbols once those understandings are strong.

An understanding of five, regardless of form, is a concept.

Counting these beans is a skill.

In putting a class together on math for preschool children, I quickly discovered that neither my students nor the literature on math for young children had a common vocabulary. The names for the basic abilities we wished children could acquire were not clear. For example, people didn’t have a clear way to describe the ability to say “15” as the quantity of those beans.

To solve the problem I had the students work together to specify the skills and concepts in arithmetic. To present the problem, I tape-recorded my son, Benj, who happened to be four years old at the time.

You can see that video below. I presented tasks for Benj to do without the naming them, so we could observe abilities and difficulties a child faces in early arithmetic. I would stop the tape after each to discuss what was happening and agree upon a common vocabulary.

You can do it this way, too, by stopping to discuss each before going on to the next, I highly recommend taking the time to do it, because you’ll see how perspectives are widely different, and, if you look at the literature, sources often use different words for what we want to see in children.

Given the picture of beans above, what name would you give to the ability to say, “Fifteen”?

Some call it counting, saying the number names, one-to-one correspondence, counting items, or counting up. I call it counting how many. It’s a skill that one develops through practice.

Watching the Video

Before you look at the list below the video, you can watch with others and stopping the video piece pausing to discuss what each ability is called, as well as what Benj is doing.

  1. If you know each ability, you can see when children do it naturally.
  2. Seeing the distinction between concepts and skills helps people recognize how children have to build their understandings themselves.
  3. You can articulate to parents and supervisors what’s important.
  4. You can better evaluate mathematics materials, video games, and toys.
  5. You can clearly distinguish assessment from education, which people often assume are similar.

I find it most intriguing to try to figure out what this four-year-old knows and is able to do, which is more fun to do when you hear other points of view. You can see more of what is happening. I see the anxiety: being on the spot, exposed, videotaped, and trying to do the very best.

Afterward, you will see that I missed having him count on and write numerals.

Arithmetic Skills and Knowledge

Download PDF Arithmetic Skills




One name you may not find elsewhere is Envisioning, the ability to state the missing quantity when a portion of a given set is hidden and all one can see is the remainder. Envisioning is the ability to do that for every combination in that set size. Others have called this “really understanding” a number (which is true, but not distinctive) or partitioning (which is an action not a mental abstraction). those names fail to describe what is built in the imagination.

Here are five shiny beans. You can separate them into sub-groups. That is partitioning, i.e., 1-4, 2-3, 3-2, 4-1, 5-0.

Envisioning is having an idea of a quantity as both items and holes. See the holes?

The “holes” are the imagined beans under the cup. Once one has the vision of five in mind, one never forgets it.

It’s like Object Permanence: the train is understood as being there, inside the tunnel. No one can give understanding to anyone. It has to be constructed uniquely by each person in their dendrites inside their brain.

Envisioning is the ability to imagine the amount withdrawn—the unseen—while experiencing the perceptual dominance of the remainder, the two big beans, right there in view. That’s the trick, because the two beans naturally attract attention, making it challenging when young.

Age and Envisioning

It is fascinating how the quantity represented in the mind corresponds to the age of the child. It’s kind of amazing; the imagining of the missing part of the set generally increases by one each year of age.

I think we all may have experienced a very young child, who is given a cookie, raising the other hand for a second one. Envisioning two. Inside the brain, somehow. Two’s have two.

I have learned to say with confidence that a child is doing well at arithmetic if they can envision the number of their age before their next birthday.

If a child can envision 3 during the year they are three years old, envision 4 during the year they are four, and envision 5 during the year they are five, I have every confidence that they will envision 6 by common school age, where envisioning to 10, the underpinnings of addition and subtraction, are ideas being manipulated and the benefits applied.

No Worries. No Hurries.

The conclusion, it seems, is that it takes a year of experience to enlarge an understanding of quantity one step at a time: two, three, four, five, six.

I have found no reason to worry about the close stragglers: envisioning one’s age minus one—that is, a four-year-old can envision 3 but not 4—I have found them doing fine. Like every other ability, some children really soar in singing or dancing or striking a ball; some really soar in arithmetic. Even though we might want our child to excel everything, not everyone has to soar.

The take-away

Envisioning gives us insight into a young child’s arithmetic competence. We can check to see if everything is progressing just fine. Elementary school performance in manipulating symbols depends upon understanding the concepts, not counting skills.

Acquiring this deep understanding by manipulating numerals following an algorithm, like the one below is what makes math hard for many children. If you have to follow a procedure without understanding the beans and holes, one might end up saying, “I’m no good at math.”

Envisioning uncovers the fundamentals pervading most aspects of arithmetic and computation the child encounters later. You might note that this item is missing from kindergarten readiness tests. I like discovering something simple like this; I also like creating opportunities where this understanding is constructed.

Group Games

Research shows this is true.  Related research from the University of Missouri.

A Focus on Numerical Age

I know it sounds simple, but I had never thought that the opportunities children most needed matched their chronological age. Five is the center for five-year-olds; two is the center for two-year-olds. Age is a rough guide for activities in arithmetic when thinking of mathematics and design in a horizontal curriculum, not a vertical one. A vertical curriculum implies that faster and higher are better than wide-ranging intellectual experiences focused on deeper understanding. It seems to me that best thing one can do for groups of children is to offer a broad spectrum of opportunities and games using the children’s age minus 1 and slowly creep them upwards to age plus 1. A reasonable expectation would be that when children approach the age of six, they are close to envisioning five. With a solid image of five-ness as items and holes, six-year-olds will easily get six.

When I learned this, all the pressure to push disappeared. It’s comforting to know that children have at least four years play and following their natural interests: from the time they know about two crackers to the time they discover the magic of six is a very long time. Six, by the way, is actually very interesting.


It seems the children who envision six, seem to quickly expand that five-plus-one to five-plus-two, since they already have mastered all the combinations of five. So, children seem to get seven, eight, and nine close to the same time. You can see that represented in items and holes 0-10.

Adding increments of one onto a deeply understood five is the basic of arithmetic and opens symbolic manipulation with our number system. Numerals then represent deep concepts, eliminating the biggest problem children have later.

You can see five-plus displayed in the beads on this abacus image. The upper beads represent five, the lower beads represent one. Quantities are represented by sliding the ones up or fives down to touch the center bar.

Here are the place values used in calculating with an abacus.

Four beans and hiding cup on the bottom rods. Five (present or not present) marked on the top rods.

What’s cool is to represent this with the fingers and thumb of one hand, like below. The quantities 0 to 9 are visible on one hand, not two, by using the thumb to represent five. If we commonly used the right hand for 0 to 9 this way, we could use the left hand for keeping track of the tens.

If all those countless hours of math educator meetings simply decided to standardize finger counting this way, envisioning might be enhanced. You can find videos on YouTube showing how much calculation can happen on two hands, the left hand representing tens and the right hand representing ones.

Symbols, the numerals, remain difficult for children to use if they have not constructed the concept first. Envisioning gives us a clue.


set8Subitizing is another way to see the importance of six. Quantities of six and below can be immediately grasped without counting them out. Above six and it gets tough. You can check this for yourself by looking at the dining chairs and the mirrors. Instead of subitizing (grasping the set as a whole), people begin to subdivide the larger set of chairs into smaller sets and put them together in their head. Many people can grasp the six-ness of the mirrors immediately, without subdivisions. Six marks the edge of subtilizing. Six, again.


The shift at six must be a brain thing, most likely evolving in concert with our hands.

I wonder if the age of six or seven, along with the edge of understanding number is related to the shift into concrete operational capabilities. A change in capabilities begins common school age in most countries. Children have reached the age where they can hold two things in their minds at once, like five plus combinations. With that foundation, symbols easily become representative tools with meaning. Maybe that’s why envisioning is essential to the rest of arithmetic.

And why all those worksheets for preschoolers should be recycled into something useful.


Although I personally resist external demands to test children, I am a big fan of assessment for my own learning. I find it a necessary step in professional development to at least partially grasp how the emergence of skills and understandings proceeds. After a systematic look for a year or two, I trust an educator to never formally assess again. He or she can simply observe. That’s being a professional. I think an essential part of early childhood educator’s preparation is to systematically check out all children on the asterisked items on the Arithmetic Skills and Knowledge document at least twice a year for two years. This study enables one to see the missing — one’s own envisioning of learning — and to see how differently skills and concepts emerge over time. People taking my course in this were required to do this at least once.

Understandings Discovered

Here is a sample of what participants discovered in checking on arithmetic skills and concepts with young children in their classrooms and child care spaces.

  • We must always keep it fun.
  • We can stop any time we need to; we can always come back later.
  • Even if we have a class of children all the same age, they are at vastly different levels; that is the way it is.
  • Assessment checks can show us what level to do our demonstrations and offer opportunities so that those who are at the “lower end” are entirely successful and feel smart. The “upper end” is always just fine.
  • Often we are surprised by what is going on that we did not see before.
  • When children are wrong or can’t do a task, we must remain authentic, present, and truthful.
  • We focus on what we can see that is successful, for that is the only thing we know; when a child is not successful, we know nothing.
  • We maintain the clear distinction between feedback (right/wrong, yes/no) which people need and judgment (good, bad) which they don’t need.
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