Advanced Counting, Addition & Subtraction

Advanced Counting-by-ones Strategies

In Chapter 4 we describe how the counting-on child would solve a task such as 8 + 5 presented with two screened collections. Children who can count-on in this way can develop (or perhaps already have developed) similar kinds of strategies for subtractive tasks. Collectively, these strategies for additive and subtractive tasks involving two screened collections are referred to as the four advanced counting-by-ones strategies and they have the following characteristics:

• They involve commencing the count from a number other than one. that is. they do not involve counting-from-one (in contrast to figurative and perceptual counting).

• They involve counting forwards or backwards by ones (in contrast to a non-count-by-ones strategy).

• They involve some means of keeping track of counting-by-ones, typically using fingers or implicit or explicit double counting.

There are four of these strategics in all, and each is typically associated with a particular kind of task.

These strategies are counting-up-from, counting-up-to, counting-back-from and counting- back-to. Each is summarized in Table 6.1 and explained below.

Counting-up-from

Counting-up-from is associated with children's solutions of additive tasks involving two screened collections, for example 8 + 5. The counting-on strategy is referred to specifically as counting-up-from. When solving 8 + 5, for example, the strategy involves counting up five from eight. Additive tasks are described in Task Group A6.1.

Counting-up-to

Counting-up-to is associated with children's solutions of missing addend tasks (a kind of subtractive task) where, for example, the child is told there are 8 counters in the first collection and 13 counters in all, and has to figure out how many are in the second collection. The counting-on strategy is referred to specifically as counting-up-to. The child starts at 8 and counts up to 13, and keeps track of the number of counts after 8, that is. 5 counts. Missing addend tasks are described in Task Group A6.2.

Counting-back-from

Counting-back-from is associated with children’s solutions of a second kind of subtractive task, that is, the removed items task. In this case the child is told how many counters are in a collection, and then how many are removed from the collection. For example, there are 13 counters under the screen, and then 5 are removed, how many are left under the screen? The child uses a counting-back strategy which is referred to specifically as counting-back-from, that is, the child starts at 13, and counts back 5 from 13, to obtain the answer 8. Removed items tasks are described in Task Group A6.3.

Counting-back-to

Counting-back-to is associated with children's solutions of a third kind of subtractive task, that is, the missing subtrahend task. In this case the child is told how many counters are in a collection and how many remain after some are removed. For example, there are 13 counters under the screen, and then some are removed and 8 remain, how many were removed? The child uses a counting-back strategy which is referred to specifically as counting-back-to, that is, the child starts at 13, and counts back to 8, and keeps track of the number of counts after 13, that is 5. Missing subtrahend tasks are described in Task Group A6.4.

Counting-forward-from-one-three-times

A common approach to the initial teaching of addition and subtraction involves giving children word problems to solve and encouraging them to use materials such as counters to solve the problems. It is important to let children use the materials to solve these problems for an extended sequence of lessons. Children typically use a strategy that we refer to as counting-forward-from-one-three-times. This strategy is used for addition or subtraction problems.

Counting-forward-from-one-three-times for addition For an addition problem the child counts out the number of counters corresponding to the first addend, then does the same thing for the second addend, and finally counts all of the counters from one.

Counting-forward-from-one-three- times for subtraction In the case of a subtraction problem, the child counts out the number of counters corresponding to the minuend, then, using the collection just counted out, counts out and removes the number of counters corresponding to the subtrahend, and finally counts the remaining counters.

A critique We are critical of this approach to instruction because in our view, it tends to perpetuate the use of what we would regard as primitive counting strategies, that is, strategies characterized by: (a) always counting with perceptual (visible) items; (b) always counting- by-ones; (c) always counting from one; and (d) always counting forwards. An additional, common characteristic of this approach is to have little regard for the relative size of the numbers that the children work with. This, too, we regard as problematic and we expand on this point below.

Screened Collections versus Word Problems

In the above sections we described four kinds of tasks, one additive and three subtractive, which can be presented using screened collections of counters. Although these tasks could alternatively be presented in word problem format rather than in a format involving screened collections, we would not advocate doing so. We believe that, at least in the initial period, the use of screened collections has benefits over the use of word problems. These benefits include that the children are provided with a consistent setting (collections of counters) which they can easily imagine (when the collections are screened). Also, the use of screened collections facilitates the development of the notion of verification, that is, children come to see that it is possible to check their answers when the collections have been unscreened. Our view is that the goal of early number instruction is for children to progress to formal arithmetic knowledge, and solving a wide range of word problems is not necessarily important for this progression. Additionally, word problems can entangle children in difficulties of reading and meaning.

Choosing Numbers for Additive and Subtractive Tasks

Because the strategies associated with these tasks (the four advanced counting-by-ones strategies described above) involve counting forward or backward, the particular choice of numbers that the teacher uses in posing such tasks is very important in our view.

Counting in the Range 2 to 5 Only

As a general rule, the number of counts the child makes using any particular strategy should be in the range 2 to 5 only. We refer to this as the count number. In terms of advancing children’s number knowledge, we think it is unproductive to have children counting long sequences of numbers, and keeping track of their counts, which is inherent in these strategies. Thus, if the instructional goal is to develop these counting strategies, then, in our view, it would not be useful to present a missing addend task such as 6 + □ = 15 (where it is understood that this is presented using screened collections or as a word problem). Similarly, a removed items task such as 22 - 13 would not be useful. Nor would an addition task such as 8 + 13 be useful if one's goal is to develop the counting-up-from strategy.

Using Advanced Counting-by-ones Strategies in the Range 20 to 100

As explained in the previous paragraph, the tasks that children solve using advanced counting- by-ones strategies should involve counts (count numbers) in the range 2 to 5 only. As long as this principle is kept in mind, these strategies can be used by children in the range 20 to 100. Thus a child might be asked to solve tasks such as the following: (a) an addition task, 38 red counters and 4 blue counters; (b) a removed items task, 37 counters are placed under a screen and 3 are removed; (c) a missing addend task, 88 red counters and some blue counters make 93 in all; and (d) a missing subtrahend task, 57 counters under a screen, some arc removed and 55 remain. Tasks such as those just described could alternatively be presented as word problems. These kinds of tasks introduce children to addition and subtraction involving numbers throughout the range 20 to 100, and help to set the scene for addition and subtraction involving two 2-digit numbers (Chapters 8 and 9).

Finger Patterns and Advanced Counting-by-ones Strategies

Children who have well-developed finger patterns for numbers in the range 1 to 5 will use their fingers to keep track, as part of the advanced counting-by-ones strategies.

Using fingers to keep track on an additive task

Thus a child might work out 8 + 5 by counting on from 8 and raising a finger for each number word from 9 onward in turn for the number words from 9 to 13. The child stops at 13 because they have raised five fingers (they recognize the finger pattern for five). Thus when they' raise the fifth finger, they' know that they have made five counts, that is from 9 to 13. In this case, it is not necessary for the child to separately count the fingers from 1 to 5. Use of finger patterns in this way is relatively sophisticated. That the child raises the five fingers sequentially (versus simultaneously) does not indicate a lack of facility- with finger patterns.

Using fingers to keep track on a missing subtrahend task

A second example of more facile use of finger patterns is: 11 counters are screened, and then some are removed and now there are only 7. How many counters were removed? The child counts back from 11 and raises a finger for each number word in turn, from 10 to 7. The child stops because they got to 7, and then looks at their finger pattern and says 'four', that is, the child can recognize the finger pattern for four, without having to count their fingers from one.

Interval- and discrete-based reasoning

When the child commences by saying 'ten' in coordination with raising one finger, we regard this as signifying a jump from 11 to 10. Similarly, saying 'nine' and raising a second finger signifies a jump from 10 to 9 and so on. Alternatively, the child might commence by saying 'eleven' in coordination with raising one finger. In this case we regard this as signifying one counter. Similarly, saying 'ten' and raising a second finger signifies a second counter and so on. In the first case the chi Id's reasoning seems to involve focusing on the interval from one number to the next. We have labeled this interval-based reasoning. In the second case the child's reasoning seems to involve focusing on each counter in turn. We have labeled this discrete-based reasoning. In either case, and as with other strategies involving counting-by-ones, children will sometimes answer one more or one less than the correct answer. Determining whether the child is using interval-based or discrete-based reasoning is an important first step in addressing this issue.

Contrasting Three Solution Strategies Involving Finger Patterns

Three children are asked to solve the additive task 5 + 4 posed with two screened collections. All three children make use of their fingers to solve the task. The first child raises five fingers on one hand sequentially, then raises four fingers on the other hand sequentially, and then counts their raised fingers from 1 to 9. The second child counts from 1 to 5 and then continues their count from 6 to 9, and raises a finger in coordination with each of the number words from six to nine. The third child counts-on from 6 to 9, and raises a finger in coordination with each of the number words from six to nine. The first child is using perceptual counting, the second is using figurative counting and the third is using counting-on. Although the first child has solved an additive task invoking two screened collections, they have done so by building perceptual replacements - their fingers replace the counters. In the case of the first child, what remains to be seen is whether they can solve an additive task when the first addend is greater than five, for example, 8 + 4. On such a task it is not feasible for the child to use their fingers to build simultaneously replacements for the two addends 8 and 4. For the third child, eight stands for having counted the first collection from 1 to 8, whereas the second child apparently needs to count from 1 to 8, in order for eight to stand for a count, these strategies are also described in Early Counting and Addition.