Structuring School Opportunities

Arithmetic Skills

Children have four years — ages 3, 4, 5, & 6 — to construct a vision of five-ness, so we can relax.

In these early years 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. When children this young are engaged in their own discoveries, together with their peers, learning arithmetic is all “Easy. Easy Easy.” as my son says in this video below.

fifteenbeansI recorded the video below when my son, Benji, was 4 1/2 years old. I wanted to record a child demonstrating these skills, understandings, and connections to symbols, so my students could see all of them in action. I didn’t put names to the tasks in the video, so the students would be challenged to describe and name each one. I would show Benji doing one and pause the tape to discuss what was happening. If you have the chance, this is an interesting challenge to undertake, because we often don’t distinguish these pieces of arithmetic for preschool children, and we definitely don’t use a common vocabulary for them. Different sources use different words for something as simple as finding out how many lima beans there are in this image.
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 and model it themselves. 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. “Wow. You’ve got a math brain.”
  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. Better able to 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. More junk than gems out there.
  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. Ethics: first do no harm.
  5. The video demonstrates assessment not teaching. Benji is not learning these skills here, I am checking on what he can do on this one day at a strange blue table. Budding educators and novice parents often confuse making a child count stuff as helping him learn. Assessment is checking — totally unlike education, which is the provision of a rich set of opportunities for learners to encounter all sorts of provocative things in a community of care. Education is the sun, soil, and water for the carrots to grow over weeks and months. Assessment is plucking one out of the ground to check on it. Grilling and drilling children, as if it were teaching them something, is toxic.

The best thing to do, I think, is to watch this with others, pausing the video after each bit to discuss what the ability is called. That’s the easy part. More difficult is coming to shared conclusions about what he knows and is able to do and the effects of his anxiety from being on the spot, videotaped by his dad. You can, of course, just follow along with either the PDF document or the listing below. 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 given size set is represented in the child’s brain as both items and holes. That is a lasting construction, which no one can give to a child. To be able to imagine the amount withdrawn while seeing only the remainder the child cup2beansmust have some kind of representation in their mind of the whole set. I think this understanding is essential.

It is fascinating to me how the quantity envisioned tracks with the age of the child. I would say children are doing very well at arithmetic if they can envision the number of their age before their next birthday. The ability seems to move along by one increment per year. If a child can envision 3 during the year they are three, 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 the envisioning up to 10, the underpinnings of addition and subtraction, are implemented. That’s the ideal, but there’s little reason to be worried; envisioning age minus one — that is, a four-year-old can envision 3 but not 4 — they’re doing fine. Like every other ability, some children really soar in singing or dancing or hitting a ball; some soar in arithmetic. Even though we might want our child to excel at singing or at envisioning, not everyone has to soar. I have found this one assessment item, envisioning, can represent progress on most aspects of arithmetic for a child under age 6. I like things simple like that. I also enjoy bringing this insight into designing experiences where this understanding is constructed.

Research shows this is true.


I know it sounds simple, but I had never thought that the challenges, games, and opportunities the children most needed centered around 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 in a horizontal curriculum, not a vertical one. The best thing one can do is offer a broad spectrum of opportunities and games focused on the children’s age minus 1 and slowly creep them upwards to age plus 1. I don’t mean to imply any limits or prohibitions on higher quantities, but when I present number games, I concentrate on the children’s age, plus and minus one. I want 100% of the children to feel “easy, easy, easy” and always be successful without hindering the children who are already understanding higher numbers. 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 prepare or push disappeared. Children have at least four years playing with numbers to discover the magic of six.


It seems the children who envision six, get seven, eight, nine, and ten close to the same time. Adding one onto a deeply understood five is the threshold to mental representations of our number system. Symbols, the numerals, will remain difficult for children to manipulate if envisioning isn’t there.

set8Subtilizing is another way to see the mental construct at work. Quantities below six can be grasped mentally at once. Above six and it changes. You can check this for yourself by looking at these images. Instead of subtilizing (grasping the set as a whole), people begin to subdivide the larger set set7into smaller sets and put them together in their head.

I have no idea what brain thing is responsible for the amazing shift at six. We have five fingers on each hand, so two hands can represent quantities to ten. We can have a firm five on one hand and the next set of five in pieces on the other. With the first five envisioned, understanding quantities beyond five combines two schemas for five. The abacus is a moving display of the two schema. It seems to be one good reason why common school age begins around the world at the age when children have reached the age where they can hold two things in their minds at once and represent them with symbols. That’s why envisioning is so essential.

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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