Arithmetic Skills

Before we get into this, we ought to check to make sure what we are thinking when we refer to arithmetic for three-to-six-year-olds, before they start common school. Three parts to this: skills (practiced expertise); concepts (abstract understandings built from experiences over time); connections with symbols (forming the symbols themselves to represent envisioned numbers which enables symbolic operations up the road).

A concept example is understanding ‘five-ness’ built somewhere, somehow in the mind, an idea of that quantity, in parts and holes, regardless of its form

A skill example is naming the quantity of the beans in the image below

Connection with symbols example is reading and writing the numeral 5 with a meaning that it represents  • • • • •  and, for us, knowing, too, that without a firm concept of five-ness deeply represented some way in the mind, the symbol may not lead to understanding operations in symbolic form (computation, etc.), with the result that symbolic operations may remain as recipes (rules-to-follow)—not deeper abstract concepts— further layers of understanding (creating the ‘I’m-no-good-at-math’ syndrome) and missing the beauty and elegance that could be found in later years.

Competence and Confidence

Preschool can make a difference, especially in building dispositions to learn and belong to a learning group. Arithmetic offers a window for us to observe children’s excitement with learning to figure things out. This exploration of arithmetic for two-to-six-year-old children presents a way for early educators to design curriculum around becoming eager, quick, and impassioned.

To begin to lead toward competence and confidence like this (a horizontal view of learning rather than a vertical ladder to climb), we first must address the skills and concepts that make for success in the game.

For example, one of those fundamental skills is being able to label the quantity in a set. Here is an image of a particular quantity of white beans. Educators need  to have a precise name order to set up opportunities for children and to respond to their successes. Do you know what ‘naming-the-quantity-of-a-set’ is called?

Arithmetic Abilities

When I taught math for preschool children, neither my students nor most curriculum materials for math for young children used a common vocabulary for describing any of the skills involved. I wanted my students to know what to look for so they can use everyday life experiences and support emergence. Like most aspects of early childhood, having precision has its benefits.

On the first day of class I asked if anyone knew what to call the ability to say “fifteen” as the quantity of beans in this image. Pretty fundamental, isn’t it, to what we are addressing here?

I occasionally interject little tasks in red in the hope readers might enjoy checking out things out for themselves. I think it’s worth taking a moment to make your own decisions.
Given the picture of beans above, what name would you give to the ability to say “Fifteen”?

Some call it counting, which doesn’t make clear that it isn’t counting one-two-three-four… that matters. How do you label the action of arriving at the last number you say when you run out of things when saying the number names? The skill is to say one number to represent the quantity of the set.

Saying the number names. One-to-one correspondence. Counting objects. Counting up. Counting quantity. All seem rather imprecise expressions of something called cardinality.

I call it counting how many, because the central ability is to state the size of the set. Cardinality, an abstract noun, has no utility here; we need a verb: skills are actions. “OK, cardinal these beans, children.”

To solve this amazingly unspoken problem, I created a video of the skills and concepts without my naming each of them. I tape-recorded my son, Benj, who happened to be four years old, going through them one at a time. You can see that video below. In class, I would pause the screen to give us time to talk about each of them. This way we could not only agree upon vocabulary but also observe an assessment of that particular skill or concept.

You can do it this way, too, by stopping to discuss each before going on to the next. If you try it yourself, you can see how perspectives are widely different. Taking it a step further, you can turn to any resource you can find in books or on the Internet and see how sources often use different words.

Demonstration Video

Before you look at the answers below the video, you’ll learn much more by watching it like my class did. With at least one other person you can watch and pause to discuss each one. I highly recommend doing that with others, because video is highly complex, much more informative than a list could ever be.

It’s fundamental.

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. Everyone is intrigued.

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

Arithmetic Skills and Knowledge

Download PDF Arithmetic Skills

Here is an example of what people encounter in arithmetic materials on the market. These numerals have unit values: 3 is three units high, 10 is ten units high 6 is six units high. A finger points to the space where a one unit 1 can lay horizontally to match the height of the 10.

Here’s an opportunity to think about buying materials for a school, rather than a home.

What might children learn from having these to manipulate? Would experience with this material enhance any of the Arithmetic Skills and  Concepts here?

Envisioning

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 name the unseen amount for every combination in that one 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 conceived 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. Where in one’s imagination are the holes?

The “holes” are the imagined beans under the cup. 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.

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

Having the vision of five in mind, for any combination, is significant. That’s when the 5 symbol becomes useful.

Age and Envisioning

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

Two’s have two. 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.

From my research of this, with many children over time, I can say with confidence that a child is doing well at arithmetic if they can envision the number of their age before their next birthday. Around the time a four-year-old turns five, they should mostly have four understood.

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 dendrites understand.

No Worries. No Hurries.

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, seems perfectly fine. Possibly they have not yet had enough experience yet. 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.

Deep Understanding

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 this one. Trying to understand what is happening without envisioning the idea of each symbol is what makes math hard for many children. Without understanding the beans and holes, one might end up saying, “I’m no good at math.”

Envisioning underlies most aspects of arithmetic and computation the child encounters later. [You might note that Envisioning is missing from standard lists and tests.]

See Group Games

Research shows this is true.  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 at the children’s age minus 1 and slowly creep them upwards to their 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.

Six

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 tens.

And, returning to items and holes, both are visible and physically represented with fingers held down or straightened. The brain has to work, so it understands by doing. Do we choose to go with old tradition or try an innovation? The choice is yours.

Think of all those hours of math meetings and what resulted. Instead, if we simply decided to standardize finger counting this way, deep understanding of number and place value could 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.

Subitizing

Subitizing 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.

Assessment

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.