Counting All and Counting On are common strategies used by young students to add two numbers. It is one of the initial strategies used by students.
Counting All is a strategy used by kindergarten and beginning first-grade students. Students who have difficulty in mathematics in later months of first grade and second grade may continue to rely on this strategy. Students who continue to use this strategy may not be able to visualize or hold the number in their mind. They look at each individual element of the number quantity. This strategy works for smaller numbers, but becomes inefficient for larger numbers and students may also make mistakes in number sequence as they count the objects. It is my belief that specifically designed opportunities to use the Counting On strategy most students will make this transition in first grade. As students in first grade develop their mathematical understanding of number and addition, teachers need to craft problems for number talks that support the transition from Counting All to Counting On.
Transitioning to Counting On
It is tempting for a teacher to show students this strategy. In fact many textbooks do just that. Then have the student practice the procedure over and over. Some resources have students use a number line in order to practice this procedure. The number line is introduced in the Next Generation CSOs in the second grade. This would not be a preferred tool in kindergarten and first grade. So the question is what do you do? One recommended visual is a double-ten frame. A visual that uses two ten frames in order to help students visualize the numbers. (A double-ten frame template may be found on this site under documents.) Once you have the template I recommend designing talks where one number is constant throughout the talk. I have attached an example of double-ten games that has two as the constant number. After students are proficient with Counting On 2, I would move to Counting On 3. Then continue with other numbers, but periodically reviewing number talks from previous days. It is important for students to answer the following questions, "How many dots do you see?" and "How do you see them?" Students will learn from each other when given the opportunity. After students have developed an understanding of counting on using the double-ten frame, I would move to using number sentences without the visual representations. I continually caution teachers not to move to the abstract to quickly. If most students are having difficulty with using only the number sentences I recommend going back to the visual representations.
Support for Students who are wrestling with the idea of Counting On
Students in each classroom develop understanding at various rates. Some students may "lag" behind the others. If you have students who continue to have difficulty transitioning to using the Counting On strategy, I would work with them in small groups. You may construct the small group to include also include students who need a little more practice with Counting On, but they could serve as models for students who may be struggling. I may choose to go back to using concrete objects with these students. For example I could use small cubes and a cup. After making two groups of objects (starting with small numbers first) I would cover one set with a cup. Students would need to remember how many I covered and then count on the objects which remain visible. This could be repeated several times with the teacher taking the lead. Later or on another day students could work with a partner on this task. The partner would play the role of the teacher. Once students begin to develop understanding of Counting On, I would transition the small group to the double-ten frame representation then the number sentence representation.
Please comment on this posting as to the helpfulness of the post and any additional questions you may have. You may also offer suggestions as to how you have helped students develop an understanding of the Counting On strategy. Remember it is about helping students develop understanding not teaching a procedure.