Dynamic counting: A suggestion for developing flexibility with counting and place value

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Abstract
This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. We present the principle of Dynamic Counting, and a set of activities related to this principle, as a teaching tool that might be useful for encouraging students to 'trust the count' and develop fluency with place value.
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 dicult 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 dicult 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 dierent 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 prociency.
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 modied 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 oer 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 dicult 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
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