In Chapter 1 we discussed the innate skill of subitizing, which is the ability to know the number of objects in a small collection without counting. If, as it seems, conceptual subitizing is the prerequisite skill for learning counting, then strengthening this skill should make learning to count easier for young students.
It may seem odd to suggest that it is possible to strengthen an innate ability. But we do this continually as we grow. Humans are born with the innate abilities to move and to speak, abilities that are strengthened through developmental learning experiences. The ability to subitize can be developed as well. Thus, preschool and kindergarten teachers should consider incorporating activities that strengthen this capability in their young students.
In searching for activities that strengthen subitizing, teachers should use cards or objects with dot patterns and avoid using manipulatives. Why? The brain is a superb pattern seeker, and we want to take advantage of this capability by getting students to form mental images of number patterns.
If students use manipulatives, they are more likely to rely on counting by ones rather than on mental imagery (Kline, 1998).
Clements (1999) suggests four guidelines that should be followed when designing activities that encourage conceptual subitizing in young students. The groups to be subitized should (1) stand alone and not be embedded in pictures; (2) be simple forms, such as groups
of circles or squares rather than pictures of animals (which could be distracting); (3) emphasize (4) have strong contrast with the background. Here and Kindergarten students:
ular arrangements that include symmetry; and are some examples of activities for preschool
Subitizing is best practiced with dot card patterns, rather than with manipulatives, to enhance imagery and eliminate counting by ones.
Figure 4.1 Cards like these, which have dots placed randomly and in patterns, help young children enhance their ability to subitize, which is determining the number of a collection without counting.
Dot patterns on cards. Draw circles on cards (or punch holes in cards for use with an overhead projector). The circles (or holes) should be arranged in geometric patterns on some cards and randomly on others. Some examples are shown in Figure 4.1 (Clements, 1999).
• One activity uses cards with randomly placed dots and asks the students to say how many dots are on a card without counting them.
• Another activity is to play a matching game. Display several cards which have the same number of dots, except one. Ask the students to say which card does not belong with the others without counting the dots.
• Select decks of cards that have zero to ten dots arranged randomly and in patterns. Give a deck to each student. Ask the students to spread the cards out in front of them. Say a number and ask students to find the matching card as fast as they can and to hold it up.
On other days, use different sets of cards with different arrangements.
• Display a card and ask students to say the number that is one more than the number of dots on the card. You can have them respond aloud, by writing down the numeral, or by holding up a numeral card. Remind them to try to avoid counting the dots.
• Place dots on a large sheet of paper or poster board in various arrangements. Point to an arrangement and ask the students to say its number as fast as possible. Each time you play this game, rotate the paper or board so that it is in a different orientation.
• Another variation is to flash one particular pattern on the overhead projector for just three seconds. The goal here is to encourage the students to think about the parts of the image.
Ask them to tell how many dots were shown and to describe what they saw. You may want to flash it a second time for three seconds to give them a chance to organize their images. That second look will be unnecessary once the students get better at recognizing patterns instantaneously. Timing is important. If you show the pattern for too long, the students will work from the picture rather than from their mental image. Showing it too briefly will not give them sufficient time to form the mental image (Kline, 1998).
Visualization. Subitizing relies heavily on visualization because the goal is to determine the number of a small group of objects with a quick visual glance and without counting.
Visualization abilities develop rapidly in young children. Thus activities that rely on visual cues enhance this development and allow students to make mental connections between patterns of objects and their numerosity.
For instance, cards displaying dot patterns in specific geometric shapes (Figure 4.2) help students to associate number and geometry by purposefully combining the two.
• Visuals also help young students see that different patterns can show the various ways a number can be partitioned, or decomposed (Figure 4.3). Through partitioning, students come to understand the idea that numbers can be broken down into other numbers. They also begin to recognize the relationship of parts to the whole. When students interpret numbers in terms of part-whole relationships, they think about numbers as made up of other numbers, and this way of thinking is the major conceptual achievement of the early school years.
Figure 4.2 These types of cards combine number with geometry and are useful in developing conceptual subitizing in young students.
Figure 4.3 Showing different arrangements of the same number of objects helps students recognize different decompositions or partitioning of a number (Clements, 1999).