Subitizing: What is it? Why teach it?

Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5(7), 400-405.

Subitizing is “instantly seeing how many.”
/ ˈsu bɪˌtaɪz / [ soo-bi-tahyz ]

Perceptual subitizing

: recognizing a number without using other mathematical processes. For example, children might “see 3” without using any learned mathematical knowledge. Perceptual subitizing accounts for some surprising
abilities of infants.

Perceptual subitizing also plays a … role is making units, or single “things,” to count.

(Alison’s note… I really can’t get my head around this idea. I need to research more about it).

Conceptual subitizing

But how is it that people see an eight-dot domino and ‘lust know” the total number? They are using the
second type of subitizing. Conceptual subitizing plays an advanced-organizing role. People who “just know” the domino’s number recognize the number pattern as a composite of parts and as a whole. They see each side of the domino as composed of four individual dots and as “one four.” They see the domino as composed of two groups of four and as “one eight.” These people are capable of viewing number and number patterns as units of units (Steffe and Cobb 1988).
Spatial patterns, such as those on dominoes, are just one kind. Other patterns are temporal and
kinesthetic, including finger patterns, rhythmic , and spatial-auditory patterns. Creating and using these patterns through conceptual subitizing help children develop abstract number and arithmetic strategies (Steffe and Cobb 1988).

Subitizing and counting

Children use counting and patterning abilities to develop conceptual subitizing. This more advanced ability to group and quantify sets quickly in turn supports their development of number sense and arithmetic abilities.

What Factors Make Conceptual Subitizing Easy or Hard?

The spatial arrangement of sets influences how difficult they are to subitize. Children usually find
rectangular arrangements easiest, followed by linear, circular, and scrambled arrangements (Beck-with and Restle 1966; Wang, Resnick, and Boozer 1971). This progression holds true for students from the primary grades to college.

Children make fewer errors for ten dots than for eight when dots are in the “domino five” arrangement but make fewer errors for eight dots when using the “domino four” arrangement.

If the arrangement does not lend itself to grouping, people of any age have more difficulty with larger sets (Brownell 1928). They also take more time with larger sets (Beckwith and Restle 1966).

Finally, textbooks often present sets that discourage subitizing. Their pictures combine many inhibiting
factors, including complex embedding, different units with poor form (e.g., birds that were not simple in design as opposed to squares), lack of symmetry, and irregular arrangements (Carper 1942; Dawson 1953). Such complexity hinders conceptual subitizing, increases errors, and encourages simple one-by-one counting.

Implications for Teaching

Everybody Counts says, “In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics” (National Research Council 1989, 58).

Conceptual subitizing and number

Many number activities can promote conceptual subitizing. One particularly rich activity is “quick
images.” When I play it with kindergarten students, I have two students stand on opposite sides of an overhead projector. One student holds a pack of cards with holes punched in them (see fig. 1 for some examples). That student places one of the cards on the overhead projector, and the other student takes it off as fast as she or he can. Then the members of the class and I race to announce the number of dots. My students are delighted that they often (and honestly!) beat me to the answer.

We play by using cards like those in the top row of figure 1, because research shows that rectangular and
dice arrangements are easiest for young children initially. At first I limit the game to small numbers. Only when students are developing conceptual subitizing do we play with more complex patterns like those in the bottom row.

Of the many worthwhile variations of the quick-image activity, some are suitable for students in any
elementary grade.
• Have students construct a quick-image arrangement with manipulatives.
• With cards like those in figure 1, play a matching game. Show several cards, all but one of which have the
same number. Ask children which card does not belong.
• Play concentration games with cards that have different arrangements for each number. For a version of
this game and other helpful activities, see Baratta-Lorton (1976).
• Give each child cards with zero through ten dots in different arrangements. Have students spread the
cards in front of them. Then announce a number. Students find the matching card as fast as possible and
hold it up. Have them use different sets of cards, with different arrangements, on different days. Later,
hold up a written numeral as their cue. Adapt other card games for use with these card sets (see Clements
and Callahan [1986]).
• Place various arrangements of dots on a large sheet of poster board. With students gathered around you,
point out one of the groups as students say its number as fast as possible. Hold the poster board in a
• different orientation each time you play.
• Challenge students to say the number that is one (later, two) more than the number on the quick image.
They might also respond by showing a numeral card or writing the numeral. Alternatively, they can find
the arrangement that matches the numeral that you show.
• Encourage students to play any of these games as a free-time or station activity

The development of imagery is another reason that these activities are valuable. Conceptual subitizing is a
component of visualization in all its forms (Markovits and Hershkowitz 1997). Children refer to mental images when they discuss their strategies. In addition, we can enhance students’ knowledge of both geometry and number by purposely combining the two. For example, play quick images with arrangements like the one in figure 2 . Older students might say, “A square has four sides, and there were two dots just on each side, and four more on the corners, so I figured twelve.”

Also play quick images that involve estimation. For example, show students arrangements that are too
large to subitize exactly. Encourage them to use subitizing in their estimation strategies. Emphasize that using good strategies and being “close” are the goals, not getting the exact number. Begin with organized geometric patterns, but include scrambled arrangements eventually. Encourage students, especially those in higher grades, to build more sophisticated strategies: from guessing, to counting as much as possible and then guessing, to comparing (“It was more than the previous one”), to grouping (“They are spread about four in each place. I circled groups of four in my head and then counted six groups. So, twenty-four!”). Students do perform better, and use more sophisticated strategies and frames of reference, after engaging in such activities (Markovits and Hershkowitz 1997). For these and for all subitizing activities, stop frequently to allow students to share their perceptions and strategies.

Because conceptual subitizing often depends on accurate enumeration skill, teachers should remedy deficiencies in counting early (Baroody 1986). Cultivate a familiarity with regular patterns by playing games that use dice or dominoes. Most important, do not take basic number competencies, such as subitizing, for granted in special populations.

Conceptual subitizing and arithmetic

Use conceptual subitizing to develop ideas about addition and subtraction. It provides an early basis for
addition, as students “see the addends and the sum as in ‘two olives and two olives make four olives’ ” (Fuson 1992, 248). A benefit of subitizing activities is that different arrangements suggest different views of that number (fig. 3).

Conceptual subitizing can also help students advance to more sophisticated addition and sub-traction. For example, a student may add by counting on one or two, solving 4 + 2 by saying “4, 5, 6,” but be unable to count on five or more, as would be required to solve 4 + 5 by counting “4-5, 6, 7, 8, 9.” Counting on two, however, gives them a way to figure out how counting on works. Later they can learn to count on with larger numbers by developing their conceptual subitizing or by learning different ways of “keeping track.”