The TIM work was initiated and organized by Hartwig Meissner and his university staff members. TIM was very active in the seventies and in the eighties. There are a lot of TIM reports and articles in national and international publications. On this web page we refer to three Lists of References:

List 1 of References (# 1-90, till 1992), these papers are not electronically available.
List 2 of References (# 91-129, 1992 - 2003), these papers are electronically available (WORD active).
List 3 of References (# 123-146, starting 2002), these papers are electronically available (just click the title).
 
In 1978 Hartwig Meissner got the funds for a sabbatical from the German Research Foundation (DFG) to study how calculators are used in mathematics education in the United States of America. He met many US experts and visited about 50 classes at about 15 locations all over the States. The following summary report describes the results from that journey. It gave a major input to the future work of the TIM group.

Meissner, H. (1978): Projekt TIM 5/12 - Taschenrechner im Mathematikunterricht fuer 5- bis 12-Jaehrige. Zentralblatt fuer Didaktik der Mathematik. Jg. 10, 1978, p. 221 – 229 (# 35 in List 1 of References) The paper reports from a visit in the United States of America to study American activities in using (simple) hand-held-calculators in elementary schools (grades K - 6, age 5 - 12). The first part of this report gives a survey about various efforts and a summary about the activities in the class room. The second part tries to structure these developments to find some guidelines for future research. The report discusses the role of the calculator concerning the concept of learning and its impact on learning mental arithmetic and paper & pencil algorithms. It distinguishes two modes of data ("analogue" and "digital") and therefore two modes of number sense ("semantical" and "syntactical"). Accordingly there are two different modes of training number sense with the calculator. Expanding these ideas the calculator can be described as a function machine to develop a "function sense". The main idea is to vary the input by guess and test to get a wanted result (see also the key word  One-Way-Principle). At the end the paper formulates recommendations for future research and finishes with five questions. The first two of them are: (1) How to conserve our paper & pencil skills in parallel to the use of calculators? (2) Which computational abilities and skills are necessary for calculator users, as well as for using the calculator as for the case that the calculator suddenly gets broken?

Till today there are no answers to these questions. Worldwide we still need a new balance between the use of paper & pencil skills on the one hand and the use of simple calculators on the other. Till today schools waste a lot of time to teach paper & pencil skills which are not used any longer then outside from schools. This is the reason that we established a Discussion Forum.

1.2  Discussion Forum:   Which shall be the future of paper & pencil skills in the primary school curriculum? Using calculators for about 30 years it now becomes extremely necessary to discuss this issue in more detail: (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 in primary schools an effective number sense and sufficient mental computational abilities despite as well as via the use of calculators? We invite you to show up at the Discussion Forum web site. The mathematics education society still waits for initiatives and for new ideas. The Forum shall concentrate on the future role of paper & pencil techniques in a modern arithmetic curriculum in primary schools. How to replace the classical algorithms and skills?

1.3  Modes of Using Calculators

On this web page we now will continue to summarize TIM experiences with simple calculators. It is obvious, the easiest use of calculators is just to press the keys along a given sequence of symbols. This syntactical use runs automatically often without any reflections about what I am doing and about what kind of a calculator result I am going to expect. The mathematical problem gets reduced into a rude skill: Press the buttons and write down the number in the display.         

Analyzing the calculator activities seen in the US we also discovered a different kind of use. There were calculator games where the students had to find an appropriate starting number to produce along given rules a specific number in the calculator display. Of course, often the first guess did not hit the target number. Thus a next guess was done. And may be even some more guesses became  necessary. The students protocolled their inputs and outputs and often looking at these tables they just intuitively got a new idea how to select a next better input number. These games became an important part in our TIM projects.

1.4  Calculator Games Using the constant facility of simple calculators we can hide operators. There are several partner games: (a) Discover my operator: Student A hides an operator, how many guesses does student B need to find the operator? (b) Big ZERO: Student A hides a subtraction operator. Student B must find an input number to get the output 0 (the big Zero). (c) Big ONE: Student A hides a division operator. Student B must find an input number to get the output 1 (the big One). Two other games center around multiplication: (d) Hit the Target: A number and a target interval are given. Find a second number so that the product of the two numbers is within the interval. If necessary, guess and test again. (e) 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.

There are several TIM reports on the use of these calculator games, for example see # 43, # 44, # 45 and # 46 in List 1 of References. Some of the games also were in the center of the PhD work from Lange. Though some of the games involve negative numbers or decimal numbers there were no difficulties at all for the children to "handle" these numbers (age about 9 years upwards).

The TIM group also analyzed more than 1000 guess-and-test protocols from Hit the Target. The protocols show that after a certain training the students develop excellent estimation skills (guessing the starting number) and a very good proportional feeling (very often less than three guesses to find a correct solution). For more details see # 68, # 72 or # 78 in List 1 of References.

In most of these calculator games the calculator was used as a tool to find an appropriate input for a problem where the "result", i. e. the target number or a target interval, already is given. The appropriate working style is to guess and test. Via these calculator games the TIM group became aware how extremely important guess and test activities might be for the development of a mathematical  understanding. And we realized, that guess and test outside from mathematics education is a quite "normal" behavior. We discovered that many people in daily life start their problem solving not by "think first" or "think logically" or "just wait a bit". Having gotten the problem often people immediately start "working", without using all of the knowledge they really have in this situation. They try, they guess and verify, they try again, they make assumptions, they even change the original problem, ...  For more details see # 54 or # 62 and also # 59 in List 1 of References.

We observed this type of behavior also in mathematics class rooms. Now, more than 20 years later, we learnt from the Dual Process Theory that this behavior is natural. Our cognition operates in two quite different modes, more details see below. Here we first present an example. The children play the calculator game Decimal Grid. In this game they have to find a path in a given grid along which they must multiply (with a calculator) given decimal numbers to get a product as small as possible. The observations show a lot of trials, a lot of intuitive behavior and they uncover a lot of mental processes. For more details see the short web page decimal grid.

1.5  One-Way-Principle We also analyzed the guess and test activities in the above calculator games mathematically. Here the problem solver has to discover properties of a mathematical relationship of the type X  ----µ--->  Y or in case of the four basic operations X  ----¤k--->  Y to find then one of the missing values (Y or X or µ or ¤k). Traditionally in mathematics education we expect Y to be the unknown variable and we then start computing syntactically. But if Y is not the unknown variable we do not work by guess and test like we do in the calculator games. In classical mathematics education we first apply the idea of reverse functions or introduce formulae or use algebraic transformations of these formulae or we use different formulae to press then (different) syntactical sequences of buttons. Observing how effectively the children played the calculator games we became convinced to use guess and test activities with calculators (and computers) systematically also to introduce regular mathematical topics and we developed an additional teaching-learning method called One-Way-Principle. A first introduction was presented in # 37 and # 42, see List 1 of References. The One-Way-Principle is an intermediate step between exploring a set of examples and discovering and describing a common structure behind these examples. In the calculator games the children explore - mainly intuitively and unconsciously - properties of additive or multiplicative structures and they get a feeling for proportional reasoning. The One-Way-Principle also implies not to switch from addition to subtraction (or vice versa) or from multiplication to division (or vice versa). Instead we have to guess the missing variable and to use then again the same key stroke sequence. That means, independent from which variables are given and which are wanted, there always is only ONE WAY to solve all problems: Always use the same (syntactical) key stroke sequence of your calculator. If necessary, you must guess the input and verify then. The goal for the learner in the guess and test work is (1)   to find a good first guess ("estimation") and (2)   to reach the given target then with only a few more guesses ("number sense" or "function sense"). We also applied the One-Way-Principle in upper grades. There is a big range of possibilities. "µ" may be a symbol for  trigonometric functions and "X" and "Y" are real numbers. Or "µ" is a symbol for a percentage function (click for more details). Or "X" symbolizes an algebraic term and "Y" the appropriate graph related to it via the "function plot software µ" (see PhD work from Mueller-Philipp).

 

1.6  Remaining independent from calculators

When we work too much and too often with calculators we may run into the danger to become dependent on the use of calculators. Thus we developed a special program to reflect that situation and to avoid such a dependence. The following is a report from the TIM project Become independent from the calculator by using the calculator.

"Become independent from the calculator by using the calculator". 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 given. 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. To "measure" number sense we developed a specific test. Also the test results indicate that the project was very successful. For the complete report see   Meissner, H. (2006): Taschenrechner in der Grundschule. mathematica didactica, 29. Jg., Heft 1, p. 5-25. Franzbecker Verlag Hildesheim, Germany

1.7  Additional Calculator Activities
Still in construction

There are several types of activities to use calculators in primary grades where the calculator is used for more than solving computation tasks (replacing mental or paper & pencil calculations). Several of them are

  o   calculators to control computations already done
  o   mathematics with arrows
  o   BINGO with calculators
  o   Upside-Down
  o   

We plan to present here a summary from the papers # 33, # 35, # 41,  # 49, #60, #76 (List 1 of References),  #109 and # 114 (List 2 of  References) and # 134, # 137 and # 139 (List 3 of References).

2.   Calculators and Computers

In this part we will report from only two investigations. Both apply the One-Way-Principle. One centers around the use of the percentage key of a simple calculator. The other summarizes the results from a PhD study on using computers to develop a more intuitive concept of functions.

2.1  Percentages

Simple calculators often have a percent key which works syntactically like we speak in our daily life: "635 + 13 % =  ..." needs the key stroke sequence

6

3

5

+

1

3

%

=

2.2   Concept of Functions

Mueller-Philipp, S. (1994). Der Funktionsbegriff im Mathematikunterricht (Dissertation, in English: The Concept of Function in Mathematics Education  -  An Analysis with Respect to Cognitive Aspects of Using Computers). Waxmann, Muenster Germany Mueller-Philipp uses the One Way Principle to bridge via guess and test the gap between "term of a function" and its related graph. She shows in her dissertation that after a training of about 8 school hours (45 min each) the about 50 students (age about 16 years) develop an impressive relational understanding (Skemp definition) of linear and quadratic functions. There are summaries in English: Meissner, H.; Mueller-Philipp, S. (1993): Teaching Functions. In: „Proceedings of the Seventeenth International Conference for the Psychology of Mathematics Education“, Vol. II, p. 89-96, Tsukuba Japan 1993 (# 92 in List 1 of References, see also # 91 or # 98).

3.   Mental Processes

3.1  Overview

Using simple calculators in arithmetic education in primary schools makes it necessary to analyze carefully the ongoing mental processes. What is number sense? How does the concept of number develop in the head of the child? What happens when in this developmental process calculators appear? We interviewed children, we went into kindergartens and we observed class room activities. There are several results, see # 47, # 48, # 55, # 56, # 57, # 60  or # 79 in List 1 of References. A very profound summary is given in the Dissertation from Lange:

3.2   Number Sense

Lange, B. (1984): Zahlbegriff und Zahlgefühl (Dissertation, in English: Concept of Numbers and Number Sense). Lit-Verlag, Muenster Germany Lange gives in her dissertation a careful analysis of how young children develop their concept of numbers. She analyzes intuitive and unconscious behaviour, she describes what we mean by number sense, and she develops a test to measure number sense. In her empirical investigation with more than 500 primary school students (grade 3, age about 9 years) the students in the experimental classes play the three calculator games Hit the Target, Big Zero, Big One. Lange shows, that the use of these calculator games does not harm the computational abilities of the students in comparison with the control group.

3.3   Vorstellungen

We also have to concentrate on a more general but related question: What does it mean to learn or to understand? (See # 59 or # 87 in List 1 of References). In case studies we observed that cognitive conflicts happen (see # 64 and # 67) and that oral or written answers from the students may have a quite different meaning for the teacher than for the students. Often the answer or the behavior of a child is a response to expectations (of an audience) and not necessarily a presentation only of an individual knowledge.

Thus we distinguish very carefully external representations (in German Darstellungen) like written or spoken answers, explanations, declarations, etc, or graphs or pictures or gestures or manipulatives, etc., from the related individual knowledge behind. These mental internal representations or individual concept images we call Vorstellungen in German. To get information about Vorstellungen we must observe and interpret Darstellungen, that means we have to analyze performances. For more details on Vorstellungen versus Darstellungen see # 125, # 130, # 131, # 135 or # 138 in List 3 of References.

The goal of mathematics education is to further the development of powerful and flexible mathematical Vorstellungen. But Vorstellungen are personal and individual and there are interesting phenomena to observe. Procepts (Tall and Gray) for example describe a certain flexibility to switch from a procedural view to quite a different conceptual view (or vice versa). Or describing a cognitive development by age there sometimes seems to be an U-shaped behavior (Strauss), according to which a mental reorganization of knowledge occurs. And we all know that sometimes there is a sudden idea, and we do not know why just this idea and why just now.

3.4   Dual Process Theory

According to the Dual Process Theory our cognition operates in two quite different modes called System 1 and System 2. We therefore distinguish two types of Vorstellungen, namely spontaneous or common-sense Vorstellungen (S1-Vorstellungen) and reflective or analytical Vorstellungen (S2-Vorstellungen). This polarity in thinking already was discussed by many other authors. E.g. Vygotzki talks about spontaneous and scientific concepts. Or Ginsburg compares informal work and written work. Or Strauss discusses a common sense knowledge vs. a cultural knowledge, which both develop quite differently and even sometimes interfere. For more details see # 143 in List 3 of References  and Leron, U.; Hazzan, O. (2006): The Rationality Debate: Application of Cognitive Psychology to Mathematics Education. Educational Studies 62/2, p. 105-126.

3.5   Technology Working Style

We now will summarize our observations when children or adults work with calculators or computers. First of all, they do much more unconscious work than they realize. It seems, that the process of learning very much is dominated by guess and test. (Obviously outside from mathematics education this knowledge is well known for example to train via simulations instead of "teaching" at real machines.)

Our observations show, that calculators and computers often intuitively are used "just to check" or "just to try". The individual interactions with the machines allow the user to act spontaneously. There are two papers which concentrate on that aspect.

Meissner, H. (2003): Constructing Mathematical Concepts with Calculators or Computers. Proceedings of CERME 3. Bellaria Italy This paper discusses theoretical aspects of learning and understanding mathematics by the use of calculators or computers. We distinguish a syntactical mode of working with these machines and a sem­anti­­­c­ mode. We discuss a dualism in concept development and reflect, if this also affects the use of calculators and computers. Especially we analyze the role of guess-and-test procedures by the use of calculators and computers (see also One-Way-Principle). You may click for the complete version (# 127 in List 3 of References, 10 pages, in English)

Meissner, H. (2008): Mental Processes: Intuitive versus Analytical. Draft Version for ICMI STUDY 17 (will appear in 2008) To work on a mathematical problem we must be aware that this will happen in parallel in two different modes where a spontaneous or intuitive thinking  (S1-Vorstellun­gen) may interfere with the analytical or reflective thinking (S2-Vorstellun­gen). S1-Vorstellungen 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 consciousness. In mathematics education there is a lack of intuitive and spontaneous ideas. To overcome this deficiency we argue to use more guess and test activities using calculators or computers. Especially we suggest the specific teaching method One-Way-Principle. Examples will be given. You may click for the complete version (# 143 in List 3 of References, 5 pages, in English)