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.

A vision of five-ness? What does that mean? We have to be clear what we are talking about. Abilities in arithmetic, in these early years, 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. A vision of five-ness is a concept, not a skill. Counting these beans is a skill.

fifteenbeansWhen my son, Benji, was 4 1/2 years old I was in my second year of teaching math for young children. I discovered the books, the students, and I had no common vocabulary. For example, few people could agree upon what actually constituted counting. First we had to work together to define the skills and concepts. To make it concrete I recorded Benji who happened to be in the middle of the age range I am addressing here. So much emerged from looking at the tape. I was really lucky that day.

I recorded the tasks without putting names to them so we could watch together both the abilities and the difficulties children face in arithmetic. In class I would show them one at at time and pause to discuss what was happening. If you have the chance to do it this way in a discussion with others, I highly recommend taking the time to do it. You’ll see how our perspectives are widely different. Educators (and textbooks) often fail to distinguish these tasks. Different sources use different words for something as simple as finding out how many lima beans are in the picture.

Here you go: What would you call the ability to say, “Fifteen”?

Five Reasons for Watching This Video

  1. If educators know what each ability is, they can see when children do it, as well as precisely model it themselves in daily life. I think seeing the range of these helps people trust the natural processes of the child’s brain and recognize how fun it is for children to figure out numbers themselves.
  2. If educators have to assess arithmetic, it’s nice to know what the abilities actually are rather than only the ones listed on the forms. Often you can see them naturally occur without making a child do them like I did in the video, so you can articulate to parents and supervisors what’s important.
  3. It’s the best filter I know for evaluating mathematics materials, video games, and toys that are marketed to the unwary. Look beyond commercial math materials to find the good stuff.
  4. This is serious business. Formal assessment, taking a child aside to present tasks like this is asserting power. Assessment puts a child at risk of failing, which they have to do in order to find an upper limit. It is one lesson in being stupid. Do no harm.
  5. The video demonstrates assessment not teaching. Benji is not learning these skills at the time he is being recorded: he learns in real life. His dad is making him do things, and he was nice enough to go along. Budding educators and novice parents often confuse making a child count stuff as helping them learn. Assessment is a teachy thing — totally unlike education, which is the provision of opportunities for encountering provocative things in a community of care. Education is more like gardening: the seeds do the changing. Education is the sun, soil, and water for carrot seeds to grow over weeks and months. Think of assessment as plucking one carrot out of the ground to check on how its coming along.

The best thing to do, I think, is to not look at the list of skills and concepts below and watch the video piece by piece with colleagues, pausing to discuss what each ability is called. Names are one thing, but most intriguing is trying to figure out what Benji knows and is able to do, because we can see the effects of his anxiety—being on the spot, videotaped by his dad. You will see that I missed one, counting on, and I did not have him write any numerals either.

Arithmetic Skills and Knowledge

Download PDF Arithmetic Skills




The names for the skills, symbols, and concepts I use are the clearest ones I know. One name you won’t find elsewhere is fivebeansEnvisioning — my name for the ability to imagine the missing quantity, in every combination, for a given set. Others have called this “really understanding” a number. Envisioning a quantity proves to me that somehow a small set is represented in the child’s brain as both items and holes. In the lower image, the imagined beans under the cup are the “holes.” To have those pictured in one’s mind is a lasting construction, which no one can give to a child. The ability to imagine the amount withdrawn—the unseen—with the perceptual dominance of the remainder in view the child cup2beansmust have some kind of representation in their mind of the whole set. You see five; now you see only two, and those two beans dominate your thinking.

I think this understanding—envisioning all the possible combinations of a small set—is the BIG ONE. It is fascinating how the quantity learning to be represented in the mind corresponds to the age of the child. The ability seems to move along by one increment per year. I have found that children are 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 would have every confidence that they will envision 6 by common school age, where envisioning to 10, the underpinnings of addition and subtraction, are tools being employed. I have 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.

Envisioning is the place to look at young children’s arithmetic competence. I have found this one assessment item, envisioning, can represent progress on most aspects of arithmetic for a child under age 6. (You might note that this item is missing from kindergarten readiness tests.) I like it when I discover something simple like this; I also like creating opportunities where this understanding is constructed.

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

Age Centrality Builds Confidence

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.



It seems the children who envision six, seem to quickly expand that five-plus to easily envision seven, eight, and nine close to the same time. 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.

You can also see the centrality of five plus one when this method is used to represent quantities 0 to 9 in this way to use fingers to represent quantities. All takes place on one hand, because five is uniquely marked by the thumb. If we did it this way, who knows how much envisioning might be enhanced.

Symbols, the numerals, will remain difficult for children to use if envisioning isn’t represented in the brain.


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 subtilizing (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. Some people can grasp the six-ness of the mirrors; others group subdivisions. Here is the edge of subtilizing. Six, again.


The amazing shift at six must be a brain thing, most likely evolving in concert with our hands. We have five fingers on each hand, so two hands can represent quantities to ten. I wonder if the centrality of age with deep understanding of number is related to the fact that common school begins in most countries when children have reached the age where they can hold two things in their minds at once, like six. With that foundation, symbols easily become thinkable. 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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