Primary School: Calculators or Paper & Pencil Techniques?
Hartwig Meissner
Westf. Wilhelms-Universitaet
Muenster, Germany
Summary
Despite many experiences with calculators in primary schools we still teach paper & pencil techniques almost as if calculators would not exist. This paper analyzes that situation and gives suggestions how to use the calculator more effectively to further number sense and to reduce paper & pencil techniques.
State of the Art
Calculators exist since more than 30 years. There were many discussions and
investigations and many interesting projects if and how to use (simple) calculators in mathematics education in primary schools (Arens e.a. 1978, Meissner
1978, Lange e.a. 1983, Meissner 1986, Shuard 1991, Fey 1992, Ruthven 1999,
De Moore e.a. 2001, Meissner 2006a, ...). And the development still is in progress1.
It is interesting to read about the advantages of using calculators and about the
additional possibilities calculators offer for mathematics education. Research
groups from all parts of the world claim that the use of calculators in primary
schools done in their projects did not harm mental arithmetic and estimation
skills. In contrary, the use of calculators can further also those skills.
But there is a discrepancy. While in daily life situations and for business purposes everybody uses a calculator and almost nobody works with paper & pencil
skills any longer we still teach and train the paper & pencil algorithms in primary schools in many countries as if calculators would not exist at all2. Obviously
emotional components are involved to avoid the use of calculators in primary
grades. Despite the successful research results parents, teachers, politicians, and
even math educators hesitate to replace essential parts of the paper & pencil curriculum by more appropriate calculator activities. Being asked they still fear that
the use of calculators will harm number sense and computation abilities of the
young children and that without the paper & pencil abilities the children might
become totally dependent on the machine.
1
In Sweden the use of simple calculators in primary schools is quite usual (Ebbing 2003),
while some Developing Countries just start to use calculators in upper grades not allowing its
use at all in primary schools (reported at the ICMI STUDY 17 conference in 2006 in Hanoi).
2
Schipper (1998) argued, that sometimes up to 50% of the time from mathematics lessons in
grades 3 and 4 is used to introduce and to train the paper & pencil algorithms.
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This discrepancy is a challenge for the mathematics education society. On the
one hand we waist a lot of time of young children to teach things they will not
need later on outside of school, on the other hand we must convince at least the
parents and the school administrations that there are valuable alternatives. The
challenge is big (cf. NCTM 2005), but I do not know reports or projects till now
which tackle that problem substantially.
In our numerous calculator projects since the seventies we have collected a lot
of experiences3. Our last project was designed around the slogan Use the calculator to become independent from it (eight classes, grade 3, age about 9 years).
The philosophy behind that project was to use the calculator in the class room
whenever it is wanted (realistic daily life situation). And of course, we then
could not follow the usual arithmetic curriculum. Instead we had to develop calculator activities which further number sense and computation abilities on the
one hand and which on the other hand also reduce the fear that the calculator use
might harm the traditional goals.
Basically there were two types of activities. The training of mental arithmetic
was done again and again in competitions with calculator groups: Who is quicker? At the beginning each pupil wanted to be in the calculator group, later on
almost nobody wanted to be there because "I am quicker in head".
The other type of activities concentrated on the development of number sense.
Playing the calculator game Hit the Target the child had to guess one of the factors of a product where the other factor already was given4. The calculator was
used to verify the guess. When the guess was not good enough the child had to
guess again. Calculator inputs and outputs were protocolled. These guess and
test tables were the only help for the children to find (intuitively) better guesses.
There also were other competition games to further the development of number
sense. The basic idea of these competitions was for both partners to guess first
and then to compare the results by the use of a calculator: Who's guess was better? Important also that each class discussed again and again the experiences
they had made (for more details see Meissner 2006b).
With these experiences in the back we now will dare to sketch a calculator curriculum for primary grades, where the introduction and training of paper & pencil abilities gets reduced dramatically. To compensate the loss of those skills we
will concentrate on developing a more powerful number sense, on a more effective training of mental arithmetic, and on the development of intuitive estimation
skills. The calculator itself will become the important tool to realize these goals.
3
For example see Meissner (1978, 2005, 2006a), Lange (1979, 1984), Lange & Meissner
(1980, 1983), Meissner & Mueller-Philipp (1993), Mueller-Philipp (1994).
4
For more details on Hit the Target see next paragraph.
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Number sense and calculators
The central question will be, how to develop number sense by the use of calculators. Number sense demands a broad range of knowledge and skills. It also includes intuitive and unconscious components: “Number sense refers to an intuitive feeling for numbers and their various uses and interpretations; an appreciation for various levels of accuracy when figuring; the ability to detect arithmetical errors, and a common sense approach to using numbers. ... Above all, number sense is characterized by a desire to make sense of numerical situations”
(Reys 1991).
For 30 years we have studied processes of learning and understanding mathematics by the use of calculators and computers. We observed (Meissner 2003),
that children - and also adults - very often develop two types of Vorstellungen5.
At the PME Working Session on Intuitive vs. Analytical Thinking in Praha 2006
I realized that these observations also can be explained in terms of the Dual Process Theory.
According to the Dual Process Theory (DPT) our cognition operates in two
quite different modes called System 1 (S1) and System 2 (S2). To work on a
mathematical problem we must be aware that this can happen in the two systems
in parallel. Then a spontaneous or intuitive thinking (S1-Vorstellungen) may
interfere with the analytical or reflective thinking (S2-Vorstellungen). S1Vorstellungen are fast and automatic and need not much working memory, but
they are very resistant against changes. To transform or to coordinate S1 experiences into appropriate and more flexible S2 experiences the Vorstellungen must
become conscious. Only then Vorstellungen can be changed or modified or
overcome. Discussions are an important tool to bring unconscious Vorstellungen
into consciousness6.
To design a new curriculum we must be aware of these mental processes, especially because there is a lack of intuitive and spontaneous ideas and activities in
the traditional teaching and learning of paper & pencil algorithms, see Table 1
on the next page.
It is interesting to see that there is an empty frame. It demonstrates that there are
almost no S1 activities (at least in Germany in grades 3 and 4) to introduce paper & pencil algorithms. It also mirrors the impression from many teachers that
their pupils do not develop a number sense for big numbers. Table 1 also indicates what would happen if we would allow an unreflected calculator use in this
5
We use the German word Vorstellungen instead of the English word representations to
avoid wrong associations. Vorstellungen are internal mental concepts and processes. External
observable representations of Vorstellungen like written or verbal answers etc. are called Darstellungen (Meissner 2002).
6
For more details on DPT and its impact on mathematical thinking see Leron & Hazzan 2006.
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curriculum. For the traditional topics the calculator just would be used spontaneously to press key after key and to accept then the number shown in the display (S1 activity).
Curriculum
Topics
System S1
stimulus-response-training
measurement numbers
daily life experiences
"automatically"
System S2
Mental arithmetic
Number space [0-100]
Estimation
Number space
[0-1.000.000]
Approximation
skillful computations
explain how and why
numbers for computing or to compare
reflecting and training
place value system, numbers become
sequences of digits
rounding & computing & rounding
discovering specific properties
before calculating
pre P&P exercises
experiencing simple computations in
decimal place value system
P&P algorithms
introduction and training
controlling computations
control the result you got
Tab. 1. Traditional introduction of paper & pencil algorithms (German curriculum)
Here the results from our calculator investigations show an alternative. The
types of problems must be changed when we work with calculators. We need
more intuitive and spontaneous activities. But most teachers or students or even
mathematicians or researchers in mathematics education often are unaware of
their spontaneous and intuitive Vorstellungen.
In the mathematics education class room often we more or less do not realize or
even ignore or suppress intuitive or spontaneous ideas. The traditional mathematics education does not emphasize unconsciously produced feelings or reactions. In mathematics education there is no space for informal pre-reflections,
for an only general or global or overall view, or for uncontrolled spontaneous
activities. Guess and test or trial and error are not considered to be a valuable
mathematical behavior in the class room. But all these components are necessary
to develop spontaneous S1-Vorstellungen.
Our experiences show that mathematical S1-Vorstellungen also can be enriched
by a systematic use of guess and test activities. So we developed a specific
teaching method called ONE-WAY-PRINCIPLE (Meissner 1979, 2003). The ONEWAY-PRINCIPLE (abbr. OWP) can be used to explore intuitively and/or consciously many functional relationships of the type

X
Y
or in case of the four
basic operations 
k
X
Y.
The basic idea of the OWP is the following. First you have to learn the syntactical sequence of the buttons you must press to get the output "Y" when input "X"
and "" (or "k") are given). That means this is by the use of a calculator or a
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computer a very easy task. The difficulties arise when X and Y are given to find
 (resp. "" or k or "k") or when Y and  (resp. "" or k or "k") are given
to find X. In upper grades in mathematics education usually we introduce then
formulae and use algebraic transformations of these formulae or we use different
formulae to press then (different) syntactical sequences of buttons7.
Here we will concentrate on the four basic operations in the primary school curriculum, that means on problems of the type "a  b = c". Here the OWP implies
not to switch from addition to subtraction (or vice versa) or from multiplication
to division (or vice versa). Instead we have to guess "a" (or "" or b or "b") to
use then again the original key stroke sequence. That means, independent which
variables are given and which are wanted, there always is ONLY ONE WAY to
solve all problems: Use the simple key stroke sequence of your calculator for the
four basic operations (S1 activity). The goal for the learner in the guess and test
work then is to find a good first guess and to reach a given target with only a
few more guesses.
Example:
There are some calculator games
which follow the OWP. Here we will
concentrate only on one example, on
Hit the Target. In this game an interval [a,b] is given and a factor k. Find
a second factor z via guess and test
that the product "zk" is in the interval [a,b]. Write a protocol of your
guesses. In the example: Find z that
z17 is in the interval [800,801].
x 17
?
[800,801]
input
display
The game trains number sense and it offers intuitive S1 experiences with multiplicative structures. Our more than 1000 guess-and-test protocols show that the
students after a certain training develop excellent estimation skills. They guess a
very good starting number and they develop an excellent proportional feeling.
Very often they need less than three guesses to find a correct solution.
There are two other calculator games which also train an intuitive feeling for the
order of magnitude in additive respectively multiplicative structures. In the calculator game "Big Zero" we hide a subtraction operator and ask "Which is the
input for getting 0 in the display? In the game "Big One" we hide a division operator and ask "Which is the input for getting 1 in the display? Discovering the
hidden operators by guess-and-test also here develops excellent (intuitive) approximation skills.
7
Examples for upper grades are given in Meissner (2003, 2008).
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A new balance
We need a curriculum for primary schools where the children on the one hand
learn to use calculators wisely and most effectively and where on the other hand
they still are able to control their calculator results mentally at any time when
necessary. And they also should be able to compute numbers up to a certain degree without a calculator.
To do the splits we must distinguish between S1-Vorstellungen and S2-Vorstellungen. We must be aware (see Table 1) that some mathematical concepts rely
more on S1-activities and others more on S2-activities. For example the concept
"size of a number" is more appropriate for the number space [0-100], while "order of magnitude" fits more for the number space [0-1.000.000]. In the number
space [0-100] almost every number is "important", we only have 101 numbers
there. But what does that mean in the number space [0-1.000.000]? To distinguish numbers here we use the S2-activity rounding.
And what does it mean to be able to control a calculator result mentally at any
time? It may be a S1-skill (intuitive reaction) and then it runs automatically
(stimulus response reactions, 1+1, 11, ...). But asking consciously to control
your result demands a second computation (S2-reaction). Thus to introduce a
new topic the following questions may become very important:
(1) How many (intuitive) S1-abilities should we develop first? (Later on the
experiences can be discussed and transferred into conscious S2-abilities).
(2) Which mathematical concepts need a more analytical and logical introduction before we start to train them intensively to get S1-skills?).8
With calculators we can build up a more powerful number sense. Hit the Target
and other calculator games may help our children to discover intuitively a lot of
properties by guess and test. Applying the OWP systematically they also can get
a feeling for additive and multiplicative structures long before they feel the need
to learn techniques9. It is not possible to sketch a new curriculum on a few pages. Thus I will summarise only some central aspects.
Calculators. We need simple calculators with the constant facility10 for the four
basic operations. Then we can hide, guess or apply operators. We also can study,
analyze and correct or add missing variables in operator tables to discover properties of the four basic operations. The calculators are allowed from grade 2
upwards almost any time, like in daily life. The main techniques to work with a
calculator are on the S1-level: Competitions, but also guess and test, experimenting, learning by own mistakes and exploring mathematical structures and
8
This is done with paper & pencil skills in the traditional curriculum.
From this point of view, why do we need techniques for 4 operations? One each for addition
and multiplication could be sufficient?
10
These calculators get "programmed" automatically to work as operators "k".  stands for
the four basic operations.
9
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relationships. To explore "a+ = c" or "+b = c" resp. "a = c" or "b = c" by
guess and test "c" should be an interval and not a specific number.
Competitions. There is a continuous rivalry between mental activities or using
the calculator. Therefore there should be competitions again and again on "Who
is quicker?" Goal: In number space [0-100] I am quicker in head.
Crucial conflict: Computing in number space [0-1.000.000]
First of all, also here we need a more effective number sense for "a  b = c".
Our experiences with Hit the Target show that there is no problem when a and b
are two digit numbers. Our students learnt to guess the first input (= order of
magnitude) very effectively. But we think this is not enough. Therefore we have
a more radical suggestion. We interpret a number as a "measurement number"
and we split it into two parts, the "measurement aspect" (only 1 or 2 significant
digits) and the related "measurement unit". O=Ones, T=Tens, H=Hundreds, ...
then also are regarded as "units" similar to mm, m or km. If necessary the number even gets split into more components , example:
(a)
3576

THTO
3576
THTO
3
5
7
6
(b)
.
T
H
T
O
THTO
3
0
57
6
or
(c)
.
T
H
T
O
or
THTO .
T
30
H or ...
54 T
36 O
Tab. 2. Splitting numbers to compute mentally
Because of the existence of calculators we now argue that we do not need paper
& pencil algorithms any longer. They can be replaced by simpler techniques:
We must compute only with a one or two digit number, which we already did in
number space [0-100]. And we have to manipulate the "units" as we did already
before.
References
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