FORUM

Future of Paper & Pencil Skills?		Zukunft Schriftliches Rechnen?
Which will be the future of the paper & pencil algorithms for the four basic operations in the primary school curriculum?		Welche Zukunft haben die Schriftlichen Rechenverfahren fuer die vier Grundrechenarten im Arithmetikunterricht der Grundschule?

	Hans Freudenthal (1973): "Wenn unser Unterrricht heute darin besteht, dass wir Kindern Dinge eintrichtern, die in einem oder zwei Jahrzehnten besser von Rechenmaschinen erledigt werden, beschwoeren wir Katastrophen herauf." (In: Mathematik als Paedagogische Aufgabe, Vol. 1, p. 61)
	Hans Freudenthal (1973): "... we will run into a catastrophe when we today teach topics which one or two decades later will be done by calculators ..." (Translation from the German text above)
	Wilhelm Schipper (1998): "... in den Klassen 3 und 4 ... [werden] manchmal bis zu 50% der gesamten Unterrichtszeit dem Einueben der Algorithmen gewidmet." (In: Grundschulzeitschrift 1998, H. 119, p. 10)
	Wilhelm Schipper (1998): "...  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." (Translation from above)
	Jim Kaput (2002): "The importance of the ability to serve as a poor imitation of a $4.95 calculator is rapidly declining"

Despite those comments we still today teach arithmetic in primary schools almost like we did 30 years ago when there were no calculators. And this situation is true world wide. Experts at the ICMI STUDY 17 conference in December 2006 in Hanoi (Vietnam) on "Digital technologies and mathematics teaching: Rethinking the terrain" discussed that phenomenon briefly: Instead of traditional paper & pencil algorithms we will need

(1) a broader variety of mental computation abilities
(2) a more effective number sense
(3) coordinated with a solid calculator curriculum

Among the specialists there were no concrete suggestions how to face the momentary discrepancy in primary schools: Simple calculators for the four basic operations dominate calculations in daily life. They are cheap and exist everywhere. But in mathematics education in primary schools calculators are forbidden in almost every country. Instead we teach paper & pencil skills, though these techniques are not practiced any longer outside from school. How to overcome this anachronism?   

The following Discussion Forum may be one possibility to collect ideas.

Discussion Forum

Future of Paper & Pencil Skills?

How to change mathematics education in primary schools with respect to the following three major aspects: (1)   Which paper & pencil skills do we still need today? (2)   How can simple calculators be integrated into the primary school curriculum to enrich and to facilitate arithmetical activities? (3)   How to develop an effective number sense and sufficient mental computational abilities despite as well as via the use of  calculators?

We need suggestions how to develop a more technology related arithmetic curriculum for primary schools. How does the use of technology change mental processes? Which will be the future role of paper & pencil techniques in a modern curriculum? The Forum asks for alternatives, alternatives for more than 100 hours of teaching old fashioned stuff. Why do we still waste so much time of young children with algorithms and skills which they will not need any longer after school?

In this Forum we will concentrate on the topics and views of the traditional arithmetic curriculum. Here we will not discuss the role of calculators in detail. There are two separate web pages which report from calculator investigations in primary schools. The one web site summarizes results from the German research group TIM (click TIM Summary). The other web site gives an overview about different calculator investigations in primary schools in different countries all over the world (click International Survey).

To develop a more technology related arithmetic curriculum we must reflect the following questions:

To answer these questions we will start with a brief survey and discuss specific aspects. Guest comments are invited then to contribute to each of the topics or to reflect additional aspects. The following topics will be discussed, please select one and click.  

1.       Traditional Topics and Possible Changes
2.       Hidden Impacts
2.1     intuitive versus analytical
2.2     guess and test
3.       Role of Calculators
4.       Number Sense
4.1     number sense in different number spaces
4.2     calculating with important numbers 
4.3     additive and multiplicative structures  
4.4     detecting arithmetical errors
5.       Mental Arithmetic
5.1     competition: mental contra calculator
5.2     stimulus response learning
5.3     who is the quickest?
5.4     developing number sense
5.5     multiplicative number sense in [100 - 1.000] and [1.000 - 10.000]  
5.6     favorite numbers
5.7     rounding and approximation
5.8     estimation
5.9     summary
6.       Paper & Pencil Techniques
7.       Decimal Numbers
8.       Summary
9.       References

Guest comments

1. Traditional Topics and Possible Changes

Topics of a traditional arithmetic curriculum	Possible changes when using calculators
Concept of numbers in different number spaces, number space [0 - 20] number space [0 - 100] number space [0 - 1.000] number space [0 - 1.000.000] Which kind of representation of numbers (Zahldarstellung in German) furthers which kind of number concept (Zahlvorstellungen in German):  counting  numbers in your environment: Keys on phone, calculator, computer, wrist watch, street numbers, traffic signs, prices or weights or … sets of elements 10×10 abacus or 10×10 number tables 10×100 strips number line digital counters and meters (speed, distance, gas, electricity, …) sequences of digits, ...	In daily life numbers have a meaning. Using technology, numbers get reduced to sequences of digits. More emphasis is necessary to balance these views. Will there be a change of what we describe with number sense?  Is there a different number sense in different number spaces? Some experts argue there will be changes, details see Number Sense below.
Mental arithmetic  skill training (addition tables, multiplication tables, stimulus response knowledge, …) applying commutative, associative, distributive law and other rules applying consciously known properties training of tricks .....	An unreflecting use of calculators might reduce mental arithmetic abilities and skills. Some experts argue there must be changes in the traditional curriculum. We must get a new balance between using calculators and mental arithmetic. Details see Mental Arithmetic below.
Comparing the “size” of numbers in different number spaces:  bigger or smaller? almost the same “size”? role of rounding  role of approximation	comparing numbers in daily life is more than comparing sequences of digits  the same is true for rounding and approximation Some experts argue there must be changes. For more details see the following paragraph.
Measuring and Estimation One part in the German arithmetic curriculum is called Sachrechnen (aspects of environmental and domestic sciences). It includes measuring (length, time, weight, etc.), the topic money, and problem solving activities (problems from real life situations like shopping, planning an excursion, constructing a bird-cage, etc.). In Sachrechnen there is a different view on numbers. A measurement number (Groesse in German) describes the size (value, magnitude, …) of an object. It consists of two parts, a quantity number and the appropriate unit like 345 km or 2830 hours or 562048 cents. The quantity number (Masszahl in German) tells us how many units we need to represent the size of that object.	For calculating big numbers we use paper & pencil skills (or calculators). But computations with measurement numbers can be different. Here we can reduce the size of a quantity number by changing the unit (i.e. 3000 m = 3 km). And then we can compute with smaller numbers and we can avoid paper & pencil techniques (or calculators). Natural numbers also can be interpreted as measurement numbers: The number itself is the quantity number and the unit is “pieces”. We could introduce additional units like “ten-pieces” or “hundred-pieces” or “thousand-pieces”, like we do when we bundle. Shorter, via bundling we can get the additional units “tens”, “hundreds”, “thousands”, etc. Using these units we also here can reduce the size of a quantity number (i.e. 3000 pieces = 3 thousands). There is the suggestion to interpret big numbers as measurement numbers to reduce paper & pencil techniques. For more details see Number Sense below.
Decimals In a traditional German primary school curriculum (grades 1 – 4, age 6 – 9 years) decimals only are used to describe a price or a length or a distance or a weight. Each of these “daily life decimals” is a pair of quantity numbers for a pair of units: 32.49 $ = 32 $ + 49 cent, 5.99 Euro = 5 Euro + 99 cent, 12.4 cm = 12 cm + 4 mm, 1.86 m = 1 m + 86 cm, 46.385 km = 46 km + 385 m, 76,450 kg = 76 kg + 450 g, 2.625 t = 2 t + 625 kg, etc. Computations with abstract decimal numbers get introduced (in Germany)  in grade 5 upwards.	According to thousands of interviews children have access (in Germany and in many other countries) to calculators in their families as much and as early as they want, especially also in their first primary school years. Ignoring then calculators and decimals in mathematics education looks a bit ivory-towered. Instead we should develop a curriculum with a technology related decimal number concept. For more details see Decimal Numbers below.

We now will concentrate on the curriculum topics from above and discuss the hidden impact which the use of calculators might have or should have in primary schools. Guest comments will be added, too.

2. Hidden Impacts

2.1 intuitive versus analytical

According to Dual Process Theories our cognition operates in two quite different modes called System 1 and System 2 (for more details see Kahneman/Frederick 2005 and Leron/Hazzan 2006). To work on a mathematical problem we must be aware that this will happen in parallel where a spontaneous or intuitive thinking (System S1) may interfere with an analytical or reflective thinking (System S2). S1 processes 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 processes must become conscious. Discussions are an important tool to bring unconscious processes into consciousness.

Distinguishing these two types of mental processes we refer also to the polarity in thinking which already was discussed before by many other authors. Vygotzki talks about spontaneous and scientific concepts, Ginsburg compares informal work and written work, or Strauss discusses a common sense knowledge vs. a cultural knowledge. Strauss (1982) especially has pointed out that these two types of knowledge are quite different by nature, that they develop quite differently, and that sometimes they interfere and conflict (“U-shaped” behavior).

We mathematics educators should be aware that each task or problem we present to a child will stimulate an immediate reaction and will recall a “subjective domain of experiences” (Subjektiver Erfahrungsbereich, cf. Bauersfeld 1983). Unconsciously the individual will select “the best” approach for solving the given problem. We then sometimes get a spontaneous S1 reaction but sometimes also a reflective manipulating or arguing. Thus the same problem might cause different problem solving behaviors.

We will give an example: How much is 15 × 19? For the children A, B and C a calculator is available, for the children D, E and F no calculators are available.

Child A immediately answers “285”. Interviewer: Sure? Child: Yes. Interviewer: Why didn’t you use the calculator? Child: I knew already.

Child B grasps the calculator and presses the keys, answer “285”. Interviewer: Sure? Child: Yes. Interviewer: Why? Child: I used the calculator.

Child C grasps the calculator and presses the keys and answers “135”. Interviewer: Sure? Child: Yes. Interviewer: Why? Child: I used the calculator. (The boy did not realize that he pressed “09” instead of “19” and there was no intuitive System S1 reaction how big the result approximately should be.)

Child D very quick: “15 × 20  gets  300 – 15  gets  285 (using the specific property 19 = 20 – 1)

Child E very quick: “15 + 9   gets  240 + 5 × 9  gets  285 (using a different technique)

Child F: I don’t know. Interviewer: Why? Child: I need paper and pencil.

2.2 guess and test

In mathematics education generally there is a lack of accepting intuitive and spontaneous ideas. Analytical and reflective thinking is dominating. To overcome these System S1 deficiencies we argue to allow more guess and test activities and activities which stimulate immediate reactions. Especially in primary schools we should integrate more the children’s common sense knowledge and we should further more the development of spontaneous S1 behavior. Creating and testing new strategies or designing own problem solving patterns also should be furthered more intensively.

Calculators can become a big help to verify or falsify assumptions. There are several calculator games which further the development of an intuitive feeling for additive or multiplicative structures. In these games basically the calculation result already is given and the child has to find an appropriate input via guess and test.

The TIM group has systematized this guess-and-test approach for using calculators and computers and has developed a specific teaching method called ONE-WAY-PRINCIPLE (Prinzip der Einbahnstrasse in German). For more details see the TIM web page. 

3. Role of Calculators

Children love calculators. Much more computations are possible without boring calculation procedures. This advantage can be used to collect more experiences on numbers and on additive and multiplicative structures, intuitively as well as consciously.

Calculators also are tools to facilitate calculations. With such a tool it becomes easier to concentrate on the mathematical topic or on the problem, regardless how difficult or complicated the numbers and the calculations are.

These tools also can be used to explore and to discover and to guess and test. Intuitive or conscious assumptions can be tested. Own or given statements can be verified or falsified.

Calculators also can be used for stimulus response learning activities (see behaviorism). After the stimulus (press the keys to enter the problem) you immediately get the response (in the display). Examples are given below.

There is one big danger when using a calculator all the time: We get used of it. Calculators are easy to work with and they do not make mistakes. A technology related arithmetic curriculum then has to reflect two aspects:

• How can we still detect a calculation error? (see Child C above)
• And what will we do when the calculator gets lost or broken?

To summarize, we must face the problem that we may get dependent on the calculator. How much independence from calculators do we still need and how much dependence is acceptable? (See also the TIM project “Use the calculator to become independent from it” on the TIM web page).

4. Number Sense

Using calculators at any time in primary schools can be accepted if and only if the children develop a solid number sense which makes them mentally independent up to a specific degree from just pressing buttons. Especially the NCTM has published some important recommendations. For example see NCTM 2000 or
http://standards.nctm.org/document/   or
http://www.nctm.org/about/position_statements/computation.htm  or
http://www.mpt.org/learningworks/teachers/numbers_alive/resources.html.

But what is number sense? In the German language there are two words with a different meaning. Zahlbegriff is the term for describing the analytical and logical aspects. With Zahlgefuehl or Zahlengefuehl the intuitive and emotional aspects are summarized. It is interesting to see that most of the German literature concentrates on Zahlbegriff only. Most authors avoid discussing aspects of Zahlgefuehl. (There is one exception, the dissertation Zahlbegriff und Zahlgefuehl from Lange 1984.)

In the English literature however there is only the one word “number sense” and it describes a more balanced view. Here the term includes both aspects, common sense and analytical 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, cited in Sowder/ Schappelle, 1994, p. 342).

Number sense not only refers to numbers but also to both, to conscious and to unconscious techniques to manipulate numbers, and it also includes a feeling about possible outcomes of these techniques. With a good number sense we roughly can predict the result of calculations, sometimes spontaneously (intuitively) and sometimes consciously (by approximating). Especially we can develop an intuitive feeling for additive and multiplicative structures. We can get a feeling for computation results like skilled painters sometimes have when they decide without any measuring how much wallpaper they will need for that room.

4.1 number sense in different number spaces

Young children have a more effective intuitive approach to compare the quantity of blocs in two given sets than adults have, cf. Fuson (1988), Lange/Meissner (1983) and others. They just have one short glance at the sets and they then decide, System S1 (see above) is dominant. Later on, when schooling begins, another part of number sense gets developed. Comparing quantities becomes less important than to determine the quantity of elements in each set separately.  

Adults have learnt techniques (in System S2) like counting, counting by tens, bundling, etc. And if the quantities get too big we change our view. We do not want any longer a precise exact number, rounded numbers will do, e.g. for the size of a swimming pool or a garbage container, for the distance between two cities or between the earth and the moon, or for the weight of an elephant or a lion, etc. Thus most of the multi digit numbers become unimportant.  These many multi digit numbers get replaced by only a few rounded numbers because in daily life we prefer rounded numbers. The result is, that not all big multi digit numbers are equally „important“.   

When estimating a size of a quantity we only use a few special rounded numbers. Putting all these favorite numbers on the number line we do not get an equidistant pattern but a pattern which looks more like a logarithmic pattern. It seems as if we determine the importance of numbers in a similar way as we perceive the intensity of light or of sounds (Weber-Fechner law). This would mean that especially in large number spaces there are only a few “important” numbers. The larger the number space is the more unimportant numbers it will have.  

Do we still need then the paper & pencil techniques for all these unimportant numbers, when we have calculators? Probably it would be wiser to concentrate on calculating with important numbers and to find strategies how to do this also mentally up to a certain degree when there is no calculator available. 

4.2 calculating with important numbers 

“Important numbers” are rounded numbers similar to measurement numbers (Groessen in German). They consist of two parts, a one or two digit quantity number (Masszahl) and a unit (“thousands”, “millions”, etc.). To calculate with rounded numbers we can separate the two parts. We must

• calculate with one or two digit numbers and  
• learn techniques what to do with the units.

Calculating with one or two digit numbers always was an important topic in arithmetic education. And when we follow the idea of “important numbers”, this topic still must remain important also when we use calculators in primary schools.

Which techniques are necessary then to manage the units without the use of calculators? Of course, the answer depends on the type of calculation. For addition and subtraction in a first step we will need for both numbers the same unit. The second step then will be to calculate with the (eventually new) quantity numbers. Here we might get a problem.  

Changing one of the given units also means to convert the related quantity number. It then may happen to get two quite different quantity numbers. One of them might be so much dominant by size that by common sense it does not make sense to add or to subtract these numbers. This problem we already have in our traditional arithmetic teaching and especially in Sachrechnen.  

For multiplication and division we must apply quite different techniques to manipulate the units. But this is not new at all. We just may remember the techniques which were important before the computer was invented. We must re-discover the rules which were necessary to use slide rules (Rechenschieber in German).

4.3 additive and multiplicative structures  

As already seen, analytically or logically in System S2, additive and multiplicative structures demand quite different techniques. But there also is a fundamental difference with respect to intuitive components of number sense. When in
“a + b = c” or in “a - b = c” resp. in
“a × b = c” or in “a ÷ b = c”  
one of the values is given and we change one of the other values, how will the third value change? How can we develop an intuitive feeling about the possible size of that third value?

Additive structures we easily can visualize via shifts along the number line. Addition and subtraction also can be interpreted as manipulations with measurement numbers (Groessen). After some training here it is reasonably easy to predict the size of the third variable.  

The representation of multiplicative structures is more difficult. Functional aspects are more dominant. Intuitively we need a proportional feeling. Either graphs or tables are necessary to visualize the hidden properties.

For producing tables or graphs a lot of computations are necessary. Here calculators or computers can help. Especially the calculator game “Hit the Target” is an excellent tool to develop a proportional feeling. In this game a number and a target interval are given. Find by guess and test a second number so that the product of the two numbers is within the interval. If necessary, guess and test again. Empirical investigations show that already 3rd graders can develop

• excellent estimation skills (guessing a good starting number) and
• proportional feeling (only a few additional guesses necessary).

For more details see Lange-Meissner (1983) or TIM - Hit the Target.

4.4 detecting arithmetical errors

There are two possibilities to detect an error, automatically because of an intuitive feeling or consciously after doing a second computation (if possible in a different approach) to control the first computation. These two possibilities are quite different by origin.  

Automatic reactions happen in System S1. They need appropriate experiences. When we expect that our students build up such an automatic error detecting we must give them plenty of opportunities to develop the necessary number sense. This means a lot of computations including guess and test are necessary to get that intuitive feeling for additive and multiplicative structures. The use of calculators will be necessary, otherwise the computations very soon will become boring and it will not be possible then to reflect the activities, consciously or intuitively.

Doing a second computation to control the first computation however is a conscious step and a decision is necessary to do this step. There are some “logical” arguments to do this control. But this method was not very successful in the traditional curriculum, maybe because consciously you have to distrust yourself. Psychologically this is a conflict with one’s individual self-confidence.

In such a situation the use of calculators is ambivalent. On the one hand calculator calculations are very easy and explicit calculation mistakes are very seldom. But on the other hand, “because the calculator does not make mistakes”, the calculator also pretends a delusive security. Though the computation result itself may be correct, the sequence of computations done might not represent a possible solution for the given problem.


5. Mental Arithmetic

Mental arithmetic abilities and an appropriate number sense are the resources to remain relatively independent from paper & pencil algorithms and from calculators. Despite the math war discussions in the USA and despite New Math arguments we claim:

• In a technology related arithmetic curriculum we need mental arithmetic skills and abilities as much and as many as possible.
• In a technology related arithmetic curriculum we need a solid number sense to remain  mentally independent from the use of technology up to a certain degree.
• The methods how to teach and to learn and to train mental arithmetic and how to develop number sense should be as attractive as possible but it is still more important that they are successful.  

As already mentioned, we will need skill training (addition tables, multiplication tables, stimulus response knowledge, etc.) as well as applying the commutative, the associative, and the distributive law and other rules or applying consciously known properties and the training of tricks. And we need more exercises which further the development of  number sense. In addition to these traditional methods we also claim to use calculators to train mental arithmetic and to develop number sense. We will give some examples, for more details see the TIM web page.

5.1 competition: mental contra calculator

The training of mental arithmetic was done again and again in competitions in the class room. There are at least two groups, a calculator group and a group to compute mentally: Who is quicker? At the beginning each pupil wanted to be in a calculator group, later on almost nobody wanted to be there because "I am quicker in head".

5.2 stimulus response learning

We train the basic facts by using the calculator. A worksheet with about 20 tasks is given. Press the calculator keys to enter the first task, but stop before using the “=” key. Compute in your head and find the answer and then press “=”. Does the display show your mental computation result? If YES write down the answer and continue with the next task. If NO do not write down any answer and continue immediately with the next task. At the end try to solve the still missing tasks with the same method. (A similar method often is used in vocabulary learning.)

5.3 who is the quickest?

Each student gets a worksheet with about 20 tasks. Who is the quickest to solve all the tasks? Using the calculator is allowed, but not obligatory.

5.4 developing number sense

Traditional mental arithmetic concentrates on computations in number space [0 – 100] or sometimes in number space [0 – 200]. And it also concentrates on exact results. But number sense includes also an intuitive feeling about numbers and an appreciation for various levels of accuracy (Reys). Thus the traditional mental arithmetic curriculum should be extended. To do mental arithmetic exercises we must expand the number space at least onto [0 – 1.000] and we need exercises which further the development of the intuitive component of number sense. We suggest to add problems where estimation and rounding and approximation are included, sometimes intuitively and sometimes consciously.  

Competitions also are attractive methods to further the development of intuitive experiences. There are several games. In a partner game the basic idea is: Both partners must guess first and then compare the results by the use of a calculator: Who's guess was better? Important also that each class discusses again and again the experiences the students have made.

5.5 multiplicative number sense in [100 - 1.000] and in [1.000 - 10.000]  

We also claim to develop an intuitive number sense for the results of multiplication problems “a × b = c” where a and b are one- or two-digit numbers. This means we must expand in a certain degree the development of a multiplicative number sense also in the number space [100 - 10.000]. One possibility to do this is playing the calculator game Hit the Target. Here we must select then for a one- or two-digit starting number the appropriate intervals (details see in 4.3).

5.6 favorite numbers  

Look for your favorite numbers. Which are the favorite numbers of your friends? Why are they favorite? Which properties do these favorite numbers have (divisibility, factorization, square numbers, famous products, many factors, easy to compute with, etc.)? Let us collect a list of favorite numbers. Which properties are necessary for a number to become “favorite”?

5.7  rounding and approximation

Solving mental arithmetic problems in a traditional arithmetic curriculum we usually concentrate on the four basic operations and we expect exact results. This limits the number space in which we do mental arithmetic. For example nobody would demand to compute “436 × 751” mentally. But when we claim neither to teach paper & pencil techniques, nor to use calculators, we at least should have a method how to find a reasonable result. In the example given we would suggest to round the numbers and to compute then mentally with the rounded numbers.

In other words, when we expect not to become too much dependent on the use of technology we need techniques which allow mental computations with “small” numbers also when the original numbers are too big for a mental computation. This means rounding and approximation should become important aspects in a technology related arithmetic curriculum.

5.8  estimation

We distinguish between estimation and approximation. Approximation is a conscious step by step process along given rules or algorithms (analytical, in System S2). It can be explained logically and can be trained systematically (mainly manipulating sequences of digits). Estimation however is an intuitive mental action (in System S1) to guess a size or a value or a magnitude of an object. Estimation qualities depend on the experiences already made. For collecting valuable experiences it is necessary to get an immediate feedback about the quality of the estimation just done.

When we regard “436 × 751” as a global entity we can interpret it also as an object. And maybe there are experts who can estimate the size of that object without being able to give some detailed information, similar to the skilled painter who without any measuring can estimate how much wallpaper he will need for that room. But we know there are experts for smaller “objects” like “43 × 76” or “38 × 52” etc. 

These specialists got their experiences by playing the calculator game Hit-the-Target again and again (see 4.3). Here for a given number and a given interval it is necessary to estimate a second number that the product of the two numbers is within the interval. In a stimulus response process the calculator gives an immediate feedback and thus facilitates collecting intuitive experiences.

The traditional arithmetic curriculum does not present exercises to develop estimation skills for “objects” like “43 × 76” or “38 × 52” etc. But in a technology related arithmetic curriculum we should use calculators to collect appropriate experiences via guess and test.  

5.9 summary

In a technology related arithmetic curriculum we need profound mental arithmetic abilities and skills and a solid number sense. The skills and abilities depend on the size of the numbers and on the type of the calculation. There also might be “mental arithmetic” problems where two or more mental steps are necessary to get a result. In this situation we also would allow writing down individual notes if wanted. But to remain mentally independent it must be possible that all mental arithmetic results can be found without the use of the traditional paper & pencil techniques and without the use of calculators or computers.

For mental arithmetic problems we suggest to accept answers on three levels of accuracy, depending on which operation must be executed in which number space:

• exact results
• rounded results (eventually several computational steps are necessary)
• estimation results (intuitively, see 5.8)

In the following tables we will summarize our demands for the levels of accuracy for mental arithmetic computations. The accuracy depends on the numbers resp. on the related number spaces. We will concentrate here only on addition and multiplication. For subtracting of more digit numbers we suggest to select the add-on-method. And we would try to avoid mental division as much as possible (except stimulus response knowing of the division tables). Instead we suggest replacing a division problem by the corresponding multiplication problem which then should be solved via guess and test.

For mental addition and for mental multiplication we expect different levels of answers, depending on the size of the numbers:

mental addition	7 [0-10]	38 [10-99]	416 [100-999]	2345 (> 999)
8 [0-10]	exact	exact	exact (if at all)	exact (if at all)
63 [10-99]	exact	exact (two steps)	rounding (if at all)	rounding (if at all)
784 [100-999]	exact (if at all)	rounding (if at all)	rounding (if at all)	rounding (if at all)
5678  (> 999)	exact (if at all)	rounding (if at all)	rounding (if at all)	rounding (if at all)

mental multiplication	7 [0-10]	38 [10-99]	416 [100-999]	2345  ( > 999)
8 [0-10]	exact (immediately)	exact (more steps)	rounding or estimation	rounding or estimation
63 [10-99]	exact (more steps)	"estimation"	rounding (if at all)	rounding (if at all)
784 [100-999]	rounding or estimation	rounding (if at all)	rounding (if at all)
5678  (> 999)	rounding or estimation	rounding (if at all)

6. Paper & Pencil Techniques

Paper & pencil techniques were developed to get a tool for computing with big numbers. Other tools were logarithms or the slide rule (Rechenschieber in German) or different types of an abacus or different types of tables or sticks and other mathematical devices.

Our modern tools for computing with big numbers are calculators or computers. Using calculators and computers nowadays is much more comfortable than practicing paper & pencil techniques. Calculators are quicker and safer to get the correct result. Thus paper & pencil techniques are not practiced any longer outside from schools.

Teaching and learning paper & pencil techniques in primary schools is combined with two problems:

• The numbers get reduced into sequences of digits (Rechenzahl in German). The original meaning of the number as a measurement number (Groesse) or a quantity number (Maßzahl) gets lost. Of course, then also the computation result  only is just a sequence of digits.
• The students get confronted with the burden to learn several different techniques to manipulate sequences of digits. These syntactical activities are quite different from the semantic activities they had practiced before. Very often a necessary understanding is missing. “Tell me what to do and I will do”. And often it is not that important if one of the digits is wrong or if a zero is missing, etc.

But still today we teach paper & pencil techniques in primary schools as if there were no calculators. To overcome this anachronism this Forum asks for alternatives. We need a shift into a technology mentality. We consciously must change our view on numbers and we consciously must accept that our life today is dominated by technology. We need a rational debate on how much dependence on and how much independence from technology our society will need.

Using calculators we still will have the problem that numbers get reduced into sequences of digits. But the “techniques” get much easier. Thus it is not very realistic to ignore or to forbid calculators in primary schools. On the contrary, all investigations with calculators in primary schools show that for the children the use of calculators is highly motivating. Thus we recommend

• to allow the use of calculators in mathematics education in primary schools whenever possible,
• to train mental arithmetic as much as possible (examples see above),
• to further the development and to train number sense, also with the calculator as a tool,
• to reflect calculator experiences in the class room as much as possible.

But there also are limitations, see the tables in 5.9 above. All children in primary schools should become able

• to add two one- or two-digit numbers without using a calculator,
• to multiply a one-digit number with a two-digit number without using a calculator,
• to give reasonable estimation results for multiplications of two two-digit numbers.

Which then will be the future role of paper & pencil techniques in a technology related arithmetic curriculum? There is no urgent need for these skills at all when the calculator is allowed almost everywhere. But to remain independent from technology up to a certain degree we still recommend developing strategies how to control the calculator result without using a calculator. How can we get a rough approximation? (May be the calculator is broken or got lost.)  

The answer is quite simple now. First we replace the big numbers by more important numbers which means that we round the big numbers to get one- or two-digit quantity numbers with adequate “units”, see 4.2 above. Then instead of computing with the big numbers we can compute (mentally) with small numbers and handle the units separately.  

Addition problems with big numbers thus get reduced into addition problems with one- or two-digit numbers. These computations we can do mentally, see table in 5.9. This means we do not need any paper & pencil algorithms for addition.

Multiplication problems with big numbers also get reduced into multiplication problems with one- or two-digit numbers. The multiplication of a one-digit number with a two-digit number we also can do mentally (see table in 5.9). The only “paper & pencil” technique we need is how to multiply two two-digit numbers to get an exact result. One suggestion might be, for example,

How to solve  46 × 83:

	×	8 tens	3 ones		×	8 tens	3 ones
Use the table	4 tens	hundreds	tens	and compute:	4 tens	32 hundreds	12 tens
	6 ones	tens	ones		6 ones	48 tens	18 ones

Step by step you get:
46 × 83  =  32 hundreds  +  (12 + 48) tens  +  18 ones  =   32 hundreds  +  61 tens  +  8 ones  =  3818


7. Decimal Numbers

When we will introduce calculators in primary schools we must be aware that the students very soon will discover also decimal numbers in the display. A technology related arithmetic curriculum cannot ignore that fact. That means such a curriculum also must sketch the first steps how, together with the calculator, the concept of decimals can be introduced in primary schools. This chapter will describe some ideas.

Already kindergarten children get curious about letters or numbers which they discover in their environment. They realize that these symbols have a name and that certain sets of these symbols form words like ANN or MAX or JOHN etc. They also identify numbers on coins and learn that certain numbers may represent different values. They learn how old they are and they can write a street number where they live or some can recite the phone number of their parents.

Thus it is not very astonishing that in all TIM calculator investigations in primary schools the students immediately started discovering their calculators: How to get a big number in the display? Which is the biggest number? Do the keys for +, -, x or : really work as I expect? And what does the key “.” (decimal point) mean? And they are happy and satisfied when they learn, that 23.5 can be interpreted as 23 cm and 5 mm, or 12.69 as 12 $ and 69 ct or 3.125 as 3 km and 125 m. And when there are more digits behind the “point”? Usually the children accept the simple answer “just ignore those digits” (which corresponds to the view from above to distinguish between important and unimportant numbers).

But the problem is more serious. In traditional arithmetic curricula from different countries decimal numbers get introduced as special fractions with denominators 10 or 100 or 1000 etc. That means fractions in those curricula must be introduced first. This situation in a technology related environment is as anachronistic as still teaching paper & pencil techniques.

And what is a decimal then? This question is more difficult to answer from curriculum planners than from primary school students. The children will give examples from decimals and perhaps some will point out that exactly one decimal point will be necessary in that sequence of digits. Do we need more for the beginning?

There should be no formal introduction of decimals when starting a technology related arithmetic curriculum. Decimals exist in daily life like letters and words. They describe certain properties and sometimes decimals also appear on a calculator display. In all these situations it really is not necessary to know that decimals are specific fractions.

If we accept that view we can discover empirically many aspects of a challenging world of decimals. There is a dominant key on the calculator to enter that world, it is the button “:” for division. Most simple calculators allow for each of the four basic operations to store an operator. One of the first calculator games then is to find via guess and test a hidden operator. For the students this is a game. But guessing and testing intuitively they learn to distinguish between additive and multiplicative structures and they get a feeling for the order of magnitude of decimal numbers. 

There are many calculator games where the students, often via guess and test, must find an appropriate input to produce the already given output number in the display. Via these guess and test activities they intuitively develop an effective number sense, for natural numbers as well as for decimals. Here we will recommend three more calculator games.

In BIG ONE one student hides a division operator and the other student then must find an input number to get the output 1 (the big One). TIM investigations show that this game can be played very successfully in grade 3 upwards.  

In Factor Finding a starting number is given and a target interval. Find an appropriate second number so that the product of the two numbers is within the interval. If you did not hit the interval, use the display number as your new starting number and find a new second number to reach still the same target interval. This game is for grade 5 upwards. It trains an intuitive percentage feeling.

In the Decimal Grid Task a geometrical net is given with a lot of many different paths. Each path consists of several steps and each step is connected with a decimal number. Select a path and multiply the decimal numbers of each step you go. Find the path with the smallest product. This game is for grade 5 upwards. Playing this game for many students it was the first time to realize that multiplication not necessarily makes bigger. For more details see Meissner xxxx.    


8. Summary

Our traditional arithmetic curriculum for primary schools must be revised. We need a more technology related arithmetic curriculum. In daily life it is quite natural to use calculators. A modern school curriculum has to face such a reality. But to follow this statement consequently we then must introduce and use calculators also in primary schools.

It is interesting to see that there is a worldwide resistance against the use of calculators in primary schools. Do teachers and parents and school administrations really believe that the teaching of paper & pencil techniques is still more valuable for our children than a modern calculator-aware curriculum? (See also Shuard 1992 or Ruthven XXXX or the calculator web page  International survey. Probably it is not only the love for the old fashioned teaching of paper & pencil techniques but the fear that an early use of calculators might harm some of the traditional goals of mathematics education.

Of course, at a first glance there are dangers. The use of calculators is that easy and simple that without an adequate curriculum almost everybody would prefer using a calculator than to compute without it. This unreflecting use will further a mental dependence on the use of technology and might endanger traditional goals of mathematics education.

But we can use the calculator also as a methodological aid

• to train mental arithmetic,
• to detect or discover number properties,
• to develop a feeling for additive versus multiplicative structures (proportionality),  
• to verify or falsify statements,
• to guess and test assumptions,  
• to explore new number spaces,  
• to detect and explore decimals phenomenologically.

To summarize, a technology related arithmetic curriculum for primary schools should  

• reduce paper & pencil techniques dramatically,
• allow the use of calculators whenever possible,
• introduce and train more intensively mental arithmetic skills and abilities (sometimes with exact results, sometimes results via rounding, also in larger number spaces),
• favor more rounding and approximation activities,  
• further more intensively the development of various aspects of number sense (especially the development of intuitive components).

9. References

For practical reasons we will refer to only one reference list. This list also will include references from the guest comments. Thus you can find the List of References at the end of the guest comments. To go there just click for References.  

Guest Comments

Everybody is invited to contribute to each of the topics or to reflect additional aspects. Please send your comments via email to Hartwig Meissner (meissne@uni-muenster.de). A summary of the inputs will be given here.

There already are some more contributions. A translation into English is in preparation. Sorry for the delay.

Es gibt schon Beitraege in Deutsch. Sie folgen hier, so bald uns auch die englische Uebersetzung vorliegt.

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After the project
Become independent from the calculator by using the calculator
(see Meissner 2006 or the TIM Summary web page) there are two more papers to discuss the future of paper & pencil techniques:

Hartwig Meissner: Arithmetikunterricht modernisieren (How to modernize arithmetic teaching) A research report given at the annual GDM Conference in Berlin 2007. There is a handout "Taschenrechner kontra Schriftliches Rechnen" (4 pages, in German) and the paper in the Proceedings (4 pages, in German). Summary: The papers identify two types of mental processes and discuss then the implications on the development of more mental arithmetic, a better number sense and more independence from the calculator.

Hartwig Meissner: Primary School - Calculators or Paper & Pencil Techniques? (Click for the complete paper in English, 8 pages) The paper analyzes the development of number sense and distinguishes two modes of thinking (spontaneous or intuitive versus analytical or reflective). We therefore observe two types of mathematical activities and two types of number sense. It is interesting to see, that the traditional curriculum separates these two types of working and thinking into two more or less independent different phases. In the lower primary grades numbers and computations in the number space [0 - 100] involve much intuitive and unconscious aspects and thus help to develop an intuitive feeling about the order of magnitude of numbers and about additive and multiplicative structures. In the upper primary grades however the numbers get bigger and the computation with those numbers gets reduced into a formal manipulation of digits (paper & pencil algorithms). The paper suggests a new balance, we must interweave these two phases and reduce the formalisms. We suggest to use calculators in grade 2 upwards mainly for two purposes, to train mental arithmetic and to further more effectively the development of an intuitive number sense in the number space [0 - 1.000]. We also suggest to rethink the role of "big" numbers. Perhaps we can split "big" numbers into two components, the "measurement aspect" (only 1 or 2 significant digits) and the related "measurement unit" (Hundreds, Thousands, ...). Computations with "big" numbers then also can be interpreted as computations with their related small measurement numbers while in addition we have to manipulate the related "units" as we already are used from traditional units (like km, m, mm, ...). Then the computations with big numbers can be done via computations with small numbers and we can rely on more mental computation abilities and a broader number sense from the number space [0 - 1.000].

Again, the floor is open for suggestions and comments. Please send your contributions to Hartwig Meissner (meissne@uni-muenster.de). A summary of your comments will be displayed here and for more detailed suggestions we can add related web addresses if wanted.