Independent learning for open society

Problems of child's psychological development. "Hot-Cold" games (for children till 7 years old). Intellectual Eye Measurer. Definitions and classification. Assessment. Computer, teacher's version. Mathematics. Statistics (for training of banking workers).



Pavel S.Pankov


Nowadays the artificial boundaries between various systems of education are being destroyed and the pedagogical science faces closely with the natural boundaries which are due to differences in language, terminology and traditions in teaching. There are two ways: to elaborate any unified world-wide system of conventions (which existed in Europe on the base of Latin in the Middle ages and is built on the basis of English now) or to develop less conventional teaching methods. Modern software promotes the second approach. The system Windows uses pictograms which can be understood by everyone. We expect that every text-book or piece of software will be examined from the standpoint of being used more widely.

While learning the essence of any subject one has to get to know the system of symbols (notations) and special terms traditionally used in it. This causes the following disadvantages:

- pupils and even some teachers confuse the content of a subject with its form, they do not recognise real life applications of its items and cannot apply their skills and knowledge;

- many persons of good ability are frightened of such a system and do not try to learn at all;

- if the state (official) language of teaching or symbols used are well known by any pupils, then those pupils have a great advantage over others.

Therefore, we suggest developing Independent Learning: ways of teaching and assessment which make minimal use of any media system not related to the content of the subject. Our proposal is distributing, with Soros Foundation, manuals for teachers, booklets with tests and computer software to enrich learning, to determine pupils' real abilities and to demonstrate that the essence of many subjects does not depend on conventional notations.

Remark 1. We do not suggest changing any existing systems of education but expect that items of this project would diversify any process of learning and make it more interesting and effective.

We hope that Independent Learning would help everyone to build and live in an Open Society.

We propose two concrete approaches with this conception: children's games of hot-cold type and intellectual eye measurer (or measurable imagination, intuition). To diversify applications of it we suggest alternative versions of fulfilling of their items.

Computer version (original pieces of software) providing for the use under the general supervision of a teacher or individually).

Teacher's version providing for active use by a teacher in a class.

Booklet version (preferably for adults) providing for the use of text-books and booklets with self-control and self-checking. It may also be used in class under the general supervision of a teacher.

Individual version. There are a lot of examples of non-conventional, active knowledge. Teachers with the best knowledge of geography could point to objects on a map, behind them, while facing the class. Although there was a map in each aeroplane, during World War II some pilots memorised the map to save seconds while in operational flight. While working air controllers must keep the position, direction and velocity of many planes in their minds. Probably, elements of this work encourage experts, parents and students to invent individual tasks for improving their own professional skills, their children's and their friends' knowledge.

Some of item listed below were approved and we hope that others would be done too.

1. Hot-Cold games (for children till 7 years old)

This approach is based on the following hypothesis: a child has an inborn sense of symmetry (in the generalised meaning) and the desire to improve.

Computer version. There is any picture on the display and the pupil may

- (* simple version) move the cursor and any pictogram within the picture;

- (** more complex version) take one of the pictograms form outside the picture and move it into the picture.

If she does it correctly, at the first attempt, then the colours of the picture become brighter and the sound of the music is more pleasant. At further attempts the colours and music mark only close approximations to the aim.

Teacher's version. A picture of approximately 10 feet * 3 feet (on a magnetic board) is presented and some (for instance, five) of the children of the group are given elements from 2 to 10 inch size (with a magnet attached). Without any instruction from Teacher each of them in turn goes to the board and fixes his/her element into place. Other children advise and make suggestions about these actions.

Further, the picture is changed and the next five of children from the group are given the elements.

Remark 2. During advising and explanation of their suggestions they are to elaborate their own language (beginning with words left, right, up, down) and Teacher is to control this process.

We give some examples for the computer version; the modification for teacher's version is obvious.


Example 1: (all kinds of) symmetry. The picture is almost symmetrical (*) except for one point or (**) except for some elements. Moving (*) the cursor to this spot or (**) any element to its appropriate place is encouraged.

Example 2: special points of figures (including centres, right angles). Some figures with marked points and (*) one figure or (**) some figures without mark are given.

Moving the cursor to (*) this point or (**) one of such points is encouraged.

Remark 3. If (**) then after marking each point the cursor returns to its expectant place until all points are marked.

Example 3: centre of symmetry of a set. Central-symmetric set of small figures is given (but its centre does not coincide with the centre of screen). Moving (*) the cursor to its centre is encouraged. (This task is more abstract than ones in Example 1; it is not convenient for teacher's version.)

These tasks differ from ones proposed by J.Piaget in absence of (verbal) demands to children.

Example 4: homothetic transformation. Some figures have their homothetic images with respect to any centre and others do not have ones within the picture (their images and some mark for the centre are elements in teacher's version and are pictograms in computer version).

We hope that reminiscent of such games would improve learning subjects at elder ages.

2. Intellectual Eye Measurer

2.1 Definitions and classification

The connection between sciences and life is usually considered as applying known methods (formulas etc.) to solve real tasks. On the contrary, we suggest the using practical experience to make teaching sciences more effective. This approach is related to the idea of advancing learning by L. Vygotsky [1] and is based on the hypothesis: much of really useful knowledge is in mind in advance although in approximate form.

Definition [7]. The problem is said to be intellectual eye measurer (or measurable imagination, intuition) if its condition is strict but the answer may be only approximate or approximate answer is permissible; using any tool (computer, paper, reference book) is forbidden; in sciences the time to answer is about 20 - 30 sec. to avoid immediate counting in head.

This approach distinguishes the following well-known fact: if any problem has a (unique) solution (object) then its solving may be considered as a translating the description of this object from the language of conditions to the language of answers.

Both a main part of condition and an answer may be presented in symbolical (name or denotation), in digital or in (more preferable) real (graphical) forms. Thus, we obtain nine possible combinations of conditions and answers:

symbolical symbolical, symbolical digital, symbolical real,..., real real.

Remark 6. We stress that an intellectual eye measurer problem of symbolical symbolical, symbolical digital, digital symbolical... types also provides any real images in mind.

Remark 7. World-wide well known pictograms may be considered as intermediate between symbolical and real types (only in conditions).

The explaining component of condition may be: verbal (text); an example (more preferable); absent (most preferable, like in above described hot-cold games but it is difficult to perform it).

Remark 8. The text is to be only necessary but not complete, for instance give the area but not give the area of the segment of a circle.

Remark 9. A computer version may also contain teaching of new nouns (only while an answer has got the excellent mark to link new terms with positive emotions), for instance

Our congratulations! You have found the area of the segment of a circle almost exactly (see Example 12 below).

Digital and real answers may also be classified as: pointwise (a number or a point); cortegewise (2..4 numbers or points); expanded (for instance, curve, graph, river, boun-dary of country).

The student may give: an approximate answer Xs itself; an interval set XI (i.e. boundaries) where the exact answer X0 is by the student's opinion surely in.

Such interval set for a number may be only a real interval (or digitally two numbers: x lower and close x upper);

for a point on a plane (a couple of numbers) it may be a real little circle (three numbers: co-ordinates x, y of centre and little radius r), a real little square (three numbers: co-ordinates x, y of centre and little side a) or a real little rectangle (four numbers: x lower and close x upper, y lower and close y upper).

An interval set for a cortegewise answer is a corresponding cortege of interval sets for pointwise answers; one for an expanded answer may be a strip (a space between two lines).

Remark 10. Here we expanded the ideas of [2] for pedagogical purposes. The interval analysis was developed originally for the automatical estimation of calculation errors; we expanded it also for computer-assisted theorem proving [3], [4].

This approach may be used both during teaching and for a quick (non-official) check of understanding sciences.

2.2 Assessment

The assessment of an approximate pointwise answer may be made in two ways: if the relative error is less than 10% then the mark is excellent; if it is between 10% and 20% then good; if it is between 20% and 40% then satisfactory else bad.

If the absolute error (the distance between the exact real answer and an approximate one) is less than 0.5 inch (at display) then the mark is excellent; if it is between 0.5 and 1.0 inch then good etc.

The assessment of an interval pointwise answer provides the following: if the student's answer XI contains the exact answer X0 then the (positive) mark is reciprocal to the width of the interval set XI (the narrower is XI the higher is the mark) otherwise the mark is very negative.

If an approximate pointwise answer is obviously impossible or interval digital answer contains obviously impossible objects then the mark is bad despite of other circumstances. (Examples of such impossibilities will be given below).

Remark 11. Because of peculiarities of interval representation this rule is not applied for interval sets of points. (For instance, if the pupil wishes to show an interval circle for a sea-shore town then s/he is to include part of a sea).

The results of such assessment may be analysed from the psychological point of view: the correlation between the real abilities of a student and self-estimations of them. If the student over-estimates his/her abilities (intervals are too narrow) then s/he sometimes obtains big minuses; on the contrary, if s/he under-estimates them (intervals are too wide) then s/he may obtain only little pluses.

Remark 12. The numbers like as 10%, 20%, 40%, 0.5 inch, 1 inch,... mentioned above are to be chosen with respect to a complexity of a task.

Remark 13. The common formula for a relative error (of positive numbers) is

E := abs(1 Xs / X0) but we suggest the following:

Elog := abs(log(Xs / X0)) because it is very close to E for Xs close to X0 but permits any positive numbers (arbitrary great or small with respect to X0 ).

The assessment of an an approximate cortegewise or expanded answer may be either by the average or the maximal of constituent pointwise deviations. Correspondingly, the positive mark of interval cortegewise or expanded answer may be reciprocal to the average or to the maximal of constituent interval sets.

2.3 Computer version

Software for intellectual eye measurer may contain:

- option choosing a language of communication;

- option choosing mode of test or learning;

- explaining text or (more preferable) demonstrating examples;

- instructions to use keys, cursor, mouse etc. (especially for moving,

narrowing and widening of intervals);

- menus; - tasks; - survey (statistics) of results and assessment.

A task may include:

- preparing of a task (by random choose);

- explaining or demonstrating setting of task;

- option help; - request of answer; - timer;

- assessment of the given answer;

- demonstration of the right (exact) answer and (optionally)

explanation of rough mistakes;

- (in the mode of learning) option for continuation.

The variety of any computer eye measurer problem may be measured by the number of randomly chosen natural N, M,... and real A, B,... quantities.

Caution: The same software fitted up for any computer may give different real sizes of figures at different displays. Hence, the doer is to foresee the necessary adaptation of any eye measurer software to real sizes of displays.

Some tasks were elaborated and used as the additional (non-official) competition at the 21st USSR National mathematical Olympiad for secondary schools held at Frunze (now Bishkek) in 1987 [6]. This approach was appreciated by the participants and leaders of teams.

2.4 Teacher's version

We describe a sketch of lesson on math or physics using teacher's version. The objectives are:

to estimate quickly whether pupils are ready to percept a new theme; to make them be interested in it;

to perturb stabilised opinions on good and bad pupils to encourage both to improve their learning.

a) Teacher says : Prepare a sheet. Close your eyes. Imagine... How long (how many...)? Write down and show me. (It takes only one or two minutes.)

Here any answer in a wide interval is right and demonstrates that a pupil has understood the preliminaries and is ready to learn the theme. A lot of nonsense answers make Teacher to correlate the prepared plan of explanation.

b) Some pupils ask: Am I right? What answer of ours is right?

Teacher answers: To get to know this we are to learn the formula (law) of.... Let us begin...

c) Teacher explains the formula (law) strictly (if possible, without appeal to evidence).

d) At the end of lesson Teacher offers to calculate the mentioned value exactly, estimates the results and declares the right (exact) answer.

e) Pupils return to their primary writings and Teacher announces the champion of intellectual eye measurer.

Remark 14. Even if an excellent pupil learns the formula (law) in advance then without thorough thinking s/he would not have any preference in such competition.

Remark 15. According to Remark 1 we advice to use this method rarely (four-five times per year) otherwise it may lower the theoretical level of teaching. It would be better if pupils themselves understand that other items of the subject taught may also be imagined without formulas.

Remark 16. Skilled teachers advise pupils to estimate whether the answer obtained by calculations is not a nonsense to avoid rough mistakes. The difference of our approach is a suggesting to imagine an answer before any calculations.

The experience of using teacher's version shows the following:

- usually one of bad pupils becomes the champion; analysis of this phenomenon reveals that, despite of forbidding, excellent pupils try to solve the task by means of nonsense formulas and calculations;

- subsequently the champion improves his learning;

- boys are better in the measurable imagination than girls;

- even at the first running of this method excellent girls feel that it threatens their rank in class and try to evade a question.

It is known that many of parents do not know modern terms learned at schools and it lowers their prestige. But they may be better in such competitions and it would demonstrate the noun knowledge of the world to their children.

2.5 Booklet version

Certainly, all examples considered below may be presented in booklets more or less satisfactorily. We distinguish examples of (... real) types. A booklet may contain two similar pages: the first, to draw answer(s) on a semi-transparent paper, and the second, with accurate answers and zones of excellent, good, satisfactory as described in section 2.2 above. After answering, the second page is to put under for check.

Now we list some examples from various subjects of primary, secondary and high school.

2.6 Mathematics

We have found the only example fulfilling the definition in 2.1 in literature: the problem Arrows [5]:

Example 5. The segment [0,1] and a little circle on it are being given at display. The pupil is to input any common fraction showing the position of the circle with error being not greater than the radius (real digital, interval with a priori given width, with verbal demand).

The examples 6-12 begin from the common eye measurer and are a step-by-step adaptation to intellectual one.

Example 6: length of a segment (real digital, with examples:

segments of 1 inch and 2 inches are given on display); variety is the length A).

Example 7: goniometry (real digital, with examples: angles of 10 degrees and 90 degrees). Our congratulations! You have measured the angle in degrees...

Example 8: length of a broken line (real digital, with examples; variety is the number N =2..4 of segments and their lengths A, B,...).

You have found the length of the broken line....

Example 9: length of a curve (real digital, with verbal demand; variety is infinite). Give the length of the curve in inches:

Example 10: area of a rectangle (real digital, with examples) variety is two sides A, B ). You have found the area of the rectangle....

Example 11: area of a triangle (real digital, with verbal demand). Give the area...:

Example 12: area of a (concave or with holes) figure by means of balancing one of varied square with it (real real, with examples; variety is infinite).

Example 13: perimeter of a triangle (real real, with examples).

You have found the perimeter of the triangle....

Example 14: pre-formula solving of equations. Teacher's version: What number being multiplied by ten is approximately equal to the sum of its square and the number two? (text digital, with verbal demand). The matter is that there are two answers (about 0.2 and about 9.8, i. e. such small that the square may be neglected and such big that the absolute term `2' can may be neglected), and pupils may discuss.

Example 15: post-formula solving of equations. Teacher's version: Solve the equation x3 + x = 20 approximately. (symbolical digital, with verbal demand).(the answer is about 2.6). Excellent pupils try to reduce it to a square equation but in vain.

Example 16: the cosine law (real digital, with verbal demand).

Teacher's version: Imagine the triangle with sides being ten and twelve inch and the angle between them being hundred twenty degrees. How long is the third (opposite) side?

Here any answer between 21 and 29 inch is right and demonstrates that the pupil has understood the preliminaries and is ready to learn the cosine law itself.

Computer version: How long is the third (opposite) side of the triangle with sides being A and B inch and the angle between them being D degrees?

Example 17: volume of a ball. What is the ratio of the volume of a ball inscribed into a cube to its volume? (symbolical digital, with verbal demand; variety is none) - only for teacher's version. The exact answer is /6 = 0.52...; the right eye measurer answer 1/2 is used to be given by engineers and skilled workers; students usually answer between 2/3 and 9/10.

Example 18: shape of a parabola. Three points of a parabola with vertical axis are given. Show the vertex of such parabola. (real real, with examples; variety is infinite).

The following two examples are elaborated with D. and J. Lugovskoys and approved at the International University of Kyrgyzstan. The first is a computer performance of the well-known image of field of slopes (directions):

Example 19: initial value problem for a differential equation of the first order: y'(t) = f(t,y(t)) (t0 t t1), y(t0) = y0. Within the strip (t0 t t1) of the (t, y)-plane a network of little arrows showing the meanings of f(t,y) is given. The task is to pass (to draw a smooth curve fitted to the arrows) from the initial point (t0, y0) until the vertical line t = t1. (real real, with text; variety is infinite).

The following was not found in literature.

Example 20: initial and boundary value problems for a differential equation of the second order: y(t) = f(t,y(t)) (t0 t t1 ), either (IVP) y(t0)=y0, y'(t0)=y01. or (BVP) y(t0)=y0, y(t1)=y1. Here the given function f does not depend on the first derivative of y and the equation can be authentically performed by the field of bends: within the strip (t0 t t1) of the (t, y)-plane the network of little arcs showing the meanings of f(t,y) is given. The task is to pass (to draw a smooth curve bending up and down accordingly to the arcs) either from the initial point (t0, y0) with the initial slope y01 until the vertical line t = t1 or from the initial point (t0, y0) till the end point (t1, y1).

2.7 Statistics (for training of banking workers)

A vast list (about 50 numbers) is given. The first question is not eye measurer yet but demands attentiveness and is an adaptation to the following.
Example 21. During 30 seconds type the least and the greatest.
Example 22: average (digital digital, with verbal demand). Type the average of these numbers. [The accurate average is to be far from the average of the least and the greatest]. If the student's answer is less than the least or greater than the greatest then it is impossible and the mark is bad.
Example 23: standard deviation. Type the standard deviation from the average for these numbers. (This problem is difficult for eye measurer and the boundary for excellent may be 20%).
2.8 Physics

Digital answers may be distinguished as relative (homogeneous to any concrete number(s) in the condition of a task) and distinct. Imagination of a distinct digital answer is difficult hence a zone for the excellent mark is to be widened.

Example 24: centre of mass. 3..10 (random) points are given on a display. The pupil is to show their centre of mass (real real, with verbal demand).

Example 25: electrical resistance of paralleled conductors.

Teacher's version: Teacher draws or shows two paralleled resistors of 20 and 10 ohm and asks: Estimate the resistance between these two points of connection (digital and real relative digital, with verbal demand).

Here any answer between 6 and 10 ohm is right and demonstrates that the pupil has understood the essence of resistance.

Example 26: rules of accelerating (decelerating) motion. Teacher's version: A stone is thrown upward vertically with the velocity 10 meters per second. What height will the stone reach? (digital distinct digital, with verbal demand).

2.9 Geography

Example 27: location of (pointwise) objects (cities, mountains etc.), location and shape of (expanded) objects (rivers, boundaries), Point (or draw) the object... on a outline-map on the display or on a semi-transparent paper (symbolical real).

Example 28: Estimate the distance between two (pointwise) objects (symbolical digital) (the list of objects is too vast to learn all mutual distances by heart).

Remark 17. The above mentioned methods are intended only for checking naturally arising knowledge of measurable imagination and do not provide a direct development of such abilities although we consider them to be very important and useful (giving such tests would develop these abilities indirectly). But we hope that after collecting some experience in this approach we would have the possibility of including developing such possibilities into learning, i.e. to enrich the content of some subjects at schools.

Remark 18. We think that after proper learning of any subject formal descriptions are to rouse images in mind similar to ones roused by real objects. We hope that the measuring imagination also provides objective assessment of fulfilling this goal.


children intellectual teacher mathematics

1. Vygotsky L.S. Problems of child's psychological development. [in Russian] - in: Selected pedagogical investigations. Moscow, 1956.

2. Moore R.E. Interval Analysis. - Englewood Cliffs. N.J.: Prentice-Hall, 1966.

3. Pankov P. Validating computations by electronic computers [in Russian]. -Ilim Publishing house, Frunze, 1978.

4. Pankov P., Bayachorova B., Yugai S. Numerical theorem proving by electronic computers and its application in various branches of mathematics. - Cybernetics, 18 (1982), no. 6.

5. Kleiman G.M. Brave New Schools: How Computers Can Change Education. - Prentice-Hall Company, 1984.

6. Pankov P., Saadabaev A. 21st All-Union mathematical school Olympiad [in Kyrgyz]. - People's Education [El agartuu],no.10 (1987).

7. Pankov P. Eye measurer problems [in Russian]. - Informatics and Education [Informatika i Obrazovanie], no.5 (1990).

8. Pankov P., Kenenbaeva G. Eye measurer methods in computer testing of knowledge in geography [in Russian]. -- Proceedings of the conference Education and Science in New Geopolitical Space. International University of Kyrgyzstan, Bishkek, 1995.

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