Authors

short

activity

Dynamic counting:

A suggestion for

developing flexibility

with counting and

place value

with Peter Sullivan and James Russo

Monash University, Vic.

It is frequently surprising to new teachers (and even those

of us with experience) when they nd that not only do

some children need to recount the group they have just

counted to be assured of the total but also that this need

seems to be resistant to intervention. Although moving

from “counting-all” to “count-on” is sometimes assumed

to be constrained by developmental factors, as teachers it

is our responsibility to continually explore methods and

pedagogies for accelerating such transitions.

Similarly, it is surprising how dicult some students

seem to nd it answering questions like 32 + 10, 95 – 10,

326 + 100, etc. Although many students in Year 1 may be-

gin to grasp place value in an elementary sense, applying

this knowledge dynamically (i.e., as a number changes),

rather than statistically (i.e., to an individual number), is

often a far slower process. Moreover, it is dicult to nd

the right experiences to help students develop uency in

exibly applying their knowledge of place value to solve

such problems. is in turn delays their ability to answer

multi-digit addition and subtraction problems using more

sophisticated partitioning and compensation strategies

(e.g., 37 + 49 = 86 because 37 + 50 = 87 and 87 – 1 = 86).

In this short article, we present the principle of Dyn-

amic Counting, and a set of activities related to this

principle, as a teaching tool that might be useful in

addressing both of these issues.

Describing Dynamic Counting

e principle of Dynamic Counting is extremely simple

but potentially powerful. e term ‘dynamic’ is intended

to convey that the totals are changing rather than static.

Imagine that students are sitting in groups of 4, with

one of them (or an adult) acting as the ‘dealer’. For

Foundation level children, the dealer progressively adds

or removes one counter from the table, while the other

group members must say the total together. As shown in

Figure 1, the total of the counters on the table might be,

progressively: 3, 4, 5, 6, 7, 6, 7, 8, 7…

e idea is that the pace of the activity illustrates to

children that it is not intended that they recount the total.

Figure 1. An Introduction to Dynamic Counting.

Rather, it is expected that they base their decision regard-

ing the new total on what the previous total was and the

action (adding or removing 1) that was done. Dynamic

Counting also seeks to assist students who see the total

number of a group as “1, 2, 3, 4, 5”, rather than “5”.

Note that this is not so much about recognising the

total quickly (subtising) but instead using the clues to

work out what is 1 more, 1 less etc. In fact, it is impor-

tant for the dealer to go beyond the numbers that the

students can recognise immediately (such as 1, 2, 3) to

the numbers for which some strategy is needed (such as

6, 7, 8, 9). e students in the group can take turns at

being dealer.

Of course, children are ready to make these connec-

tions at dierent stages. However, through having mixed

ability groups, the students who are not quite ready to

‘trust the count’ and count dynamically are still hearing

the numbers articulated as they see the corresponding

total change. In other words, participating in Dynamic

Counting can create a valuable foundational experience

for students, even if they are not always able to keep

track of the total.

Dynamic Counting is also ideal for one on one interac-

tions (such as can be done by parents, or teaching aides)

in that the pace can be adjusted and ne-tuned to suit

an individual child’s current counting prociency.

38 APMC 22(3) 2017

Even in small groups with an adult (or a peer tutor) the

group members can take turns at saying the progressive

total (rather than saying the total together) in order

to better monitor individual learning. In addition,

Dynamic Counting can also be done as a whole class

activity (e.g., as a number uency ‘warm-up’ presented

on an Electronic Whiteboard) with the class chanting

the changing total number of objects together.

Extending Dynamic Counting

Once children are comfortable with the dynamic count-

ing process, it can be extended such that the dealer adds

or removes one or two counters, rather than simply one

counter. In this more advanced version of Dynamic

Counting (see Figure 2), the total number of counters

on the table might be, progressively: 2, 4, 3, 5, 6, 7, 5,

6, 8…

Beyond counting objects, the concept of Dynamic

Counting can be modied to focus on symbolic number

representation. For example, the dealer may choose a

pre-determined rule (e.g., one more than, two less than)

and then display a sequence of single digits to which

students have to apply the rule. For example, if the rule

was ‘one less than’, students would chant the following

numbers when confronted with the digits displayed in

Figure 3: “6, 1, 5, 2, 8, 1, 3, 2, 4”.

is variation has the advantage of connecting the

number word with the associated symbols through

the same dynamic approach. Additional variations on

how the Dynamic Counting concept can be used when

working with older children are outlined below.

Year 1: e dealer places 10 counters on a tens frame (or

base 10 ‘long’) on the table. Additional counters

are then added and subtracted to dynamically

explore the numbers between 11 and 19. For

example: 11, 12, 13, 12, 13, 14, 15, 16, 15…

(see Figure 4). e purpose of this variant is

to oer students the experience that we do not

need to recount the 10, nor do we even need

to recount the additional ‘ones’, in order to

recognise a teen number.

Year 2: e dealer adds or removes either a group of

10 (base 10 long or counter labelled ‘10’) or a

single unit to begin to connect the process of

Dynamic Counting to exibly applying place

value concepts. For example: 11, 21, 31, 32,

22, 23, 33, 43, 42… (see Figure 5).

Middle primary years: e dealer adds or removes

either a group of 100 (e.g., base 10 ‘ats’; coun-

ters labelled 100), a group of 10, or a single unit.

For example: 101, 201, 211, 111, 121, 122,

132, 232, 222… (see Figure 6).

In our experience, Dynamic Counting is a process

which students at all levels nd dicult at rst, however

rapidly improve with exposure and practice. We believe

it has great promise for moving young students (e.g.,

Kindergarten, Foundation, Year 1) on from ‘counting

all’, and deepening older students (e.g., Year 1 to Year

5) understanding of place value and capacity to work

exibly with multi-digit numbers.

Figure 2. Extending Dynamic

Counting to 1 or 2 counters.

Figure 3. Linking Dynamic Counting to symbolic representation.

Figure 4. Dynamically counting

the teen numbers.

Figure 5. Dynamically counting

the numbers between 10 and 100.

Figure 6. Dynamically counting the

numbers between 100 and 1000.

39APMC 22(3) 2017