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lessons must be reformed 31 05 2010 orginal article TVN Uwaga Krzysztof Cywiński 11 03 2010 Zagadka Gazeta Wyborcza Pomysł na firmę Gazeta Wyborcza Pomysł na firmę Kronika Krakowska Children s skills under a magnifying glass 22 09 2010 orginal article Nowiny Gliwickie Scythes and razors There is something about teachers orginal article Dziennik Zachodni Everyone is guilty 07 12 2010 orginal article Dziennik Zachodni Mathematics without useless words 12

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x 2 should be determined and are unable to transform a formula and therefore rearrange the equation to obtain a trivial basic form that is easy to solve In fact only equations in a non systematic nontrivial form occur in school mathematical education above all in order to develop the elementary ability to rearrange formulas and the algebraic skill in transformations The history of mathematics proves that such algorithm could be partially formulated only in case of the trivial basic form of linear and quadratic equations The procedure of solving such equations is fragmentary due to the necessity of their occurrence in the trivial systematic form as linear equations ax b 0 where solution has the form x b a It is worth emphasising though that the sequence of necessary transposition is insignificant in this case therefore even when solving linear equations in a trivial form we are unable to explain why certain transpositions are made we may first divide then subtract and vice versa or first subtract and then divide to get the solution The result will be the same Solving a quadratic equation rearranged to a trivial basic form of ax2 bx c 0 may give one two or no results On the basis of the results of Galois and Abel it has even been proven that in case of 5th and higher degree equations even in case of trivial notation basic method does not exist while procedures of rearranging polynomial equations of all degrees to the trivial basic form of ax 2 bx c 0 może dawać odpowiednio jedno lub dwa rozwiązania bądź ich nie posiadać Dzięki wynikom Galois i Abela wykazano nawet że metoda ogólna dla równań stopnia 5 i wyższych nawet dla trywialnych postaci ich zapisu nie istnieje a procedury porządkowania równań wielomianowych wszystkich stopni do trywialnej postaci ogólnej ax n bx n 1 c 0 have not been developed In case of the latter it refers to the interest of narrow groups of specialists while in case of the algorithm of rearranging linear equations and quadratic equations to the trivial basic form which simplifies solving the issue is fundamental for the education therefore for education of future scientists and also according to C F Gauss for the reputation of the mathematics itself Without the ability to rearrange equations to a trivial form thus to solve them even if it would be done intuitively as it has been so far due to the mathematisation of the everyday life practicing science even reproductively is impossible Many scientific disciplines not only mathematical such as algebra could not have occurred Consequently mathematical education without this skill at the stage of understanding the problem should end only with addition multiplication involution subtraction division and extraction of roots Why The theorem of finding solutions of linear equations thus the ability to rearrange formulas is fundamental in relation to the Thales theorem the Pythagorean theorem and other tools of the mathematics Therefore without knowing it and without the ability to use it at least by intuition teaching geometry algebra logarithms and trigonometry as well as teaching physics chemistry and solving many significant technical issues would be impossible Notice that different approach than the binary approach supported by logic enables practicing science according to the following principle scientific truths exist but discovering them is the result of original creative actions and scientific research often irrespective of the resolutions concerning the existence of such truths II The theorem of the algorithm of solving linear equations in the form of ax b 0 The Pythagorean approach to science The world is a number implicitly the mathematics probably contributed to the issue with the following approach equations exist and people only discover them This in turn resulted in a limitation the abovementioned astonishing heuristic inertia However many science historians tend to support the claim that the Hindus were first to solve equations VIII the achievements of Arab scholars made in this field over one thousand years ago stimulated mathematics and also whole science to develop which is very significant VIII In 4 A P Juszkiewicz presents the achievements of the Hindu and the Arab mathematicians in this field while in 5 G Ifrah claims that those achievements were appropriated by mathematics historians of other cultures and demands returning to proper proportions in historical reports concerning the issue Over the centuries a trivial form of solution of the linear equation ax b 0 has been formulated and proven x b a Also proper tools to be used to get this solution have been specified the method of transposition IX whereas nobody has been able or has tried to specify an algorithm for solving such equations when their form is non systematic nontrivial just as in case of And such problems in particular develop the skills of solving equations and transforming formulas in physics or chemistry during the school education Thus it was impossible to specify a solution of a problem that is fundamental in mathematics X what should be the sequence of identity operations transpositions and how many of them should be made and are enough to be made in order to get a trivial general form of a simplest equation that is the linear equation ax b 0 IX X Renowned publishers as in 6 7 give only general theorems concerning possible transpositions in order to get a solution of a linear equation we may add any number to both sides of such equation or we may subtract any number from both sides of the equation or we may multiply or divide them by any number As there is no rule all mathematics handbooks lack the recipe telling which transpositions should be made and in which sequence This deficiency referred to as the procedural unknown by G Polya in 3 is the fundamental and evident reason for which the process of school mathematics education is perceived to be very hard and require many years of laborious practicing often doing exercises which can hardly be understood In certain simplification we may say that the abovementioned approach in the scientific and educational practice results only in a combinative and intuitive skill in solving linear equations denoted in a nontrivial form Even most outstanding mathematicians scientists solve linear equations according to the following principle I do this transposition and not another not because I know the rule determining the sequence of necessary steps to be taken in order to find the solution but because it is the way it should be done This leads to an awkward situation if a student asks why we are doing this particular transposition normally we respond bo tak się robi w matematyce because it is done this way in maths In mathematics which is deductive in its grounds through connecting results with reasons A school mathematics teacher has only their own intuition of solving equations Their task is to develop such intuition in students which in the aspect of universality of teaching must imply similarly universal problems The change in the attitude mathematics science is created by scholars irrespective of its existence allows to formulate a rule which gives mathematicians and students a tool in form of the axiom of selecting a specific transposition in the equation Theorem Doing identity operations transpositions of number a compared to letter x i e x a we may create any equation with one solution XI Reverse order of these operations gives the initial equality Transpositions that lead to a solution of the equation are made by means of elimination of redundant factors in the order reverse to the rule of the sequence of arithmetic operations doing inverse operations A factor present in a sum is eliminated by the transposition of subtraction A factor present in a difference is eliminated by the transposition of summation A factor present in a product is eliminated by the transposition of division A factor present in the denominator of the quotient is eliminated by the transposition of multiplication A factor present in the power with the exponent of 1 converse is eliminated by the transposition of raising both sides of the equation to the power of 1 XI In order to avoid unnecessary digressions the following provision must be made the article concerns equations with one unknown x occurring in the first power which may repeat within the equation Furthermore constructing equations is not an innovative process In 5 A P Juszkiewicz mentions numerous Hindu scholars who described the procedure of arranging equations for a specific task and then specified a rule of finding a solution Prtudakaswami BhaskaraII Narajana A P Juszkiewicz emphasises similarity of their methods to a detailed recipe for arranging equations when solving problems given almost a thousand years later by R Descartes in his Geometry 1637 As a result further reasoning has not resulted in formulating a general rule providing a proven theorem That is why no such theorem can be found in the handbooks Author s note To illustrate the didactic efficiency of the presented algorithm forcing the level of understanding of a problem let s analyse the given theorem using a particular example Compare any real number a to letter x to get the equality x a np x 2 According to the theorems on transposition in linear equations we may add any real number b for instance 3 to both sides of the equation then multiply both sides of the equation by any real number c for instance 4 x a b x 2 3 x b a b c x 3 5 4 c x b c a b 4 x 3 20 Performing transposition of both sides of the obtained equation in the reverse order that is eliminating factors in the order reverse to the rule of the sequence of arithmetic operations we get the initial equality of number a and letter x thus the solution of the equation the equality of number 2 and letter x c x b c a b c 4 x 3 20 4 x b a b b x 3 5 3 x a x 2 A question arises why hasn t the cited theorem been formulated up to now if it is so trivial and elementarily simple The answer must be related among others with the rigour of the methods used by the mathematicians in their practice Well if a new theorem is formulated in mathematics usually a counter example is enough to question its rightness Therefore sometimes a new theorem is formulated which is deemed right except for the given counter example if of course it is acceptable However if there are more counter examples then the theorem is given up For certain reasons more attention is paid to questioning the theorem than the counter example The discussion on the proposed use of the algorithm in school practice in the third part of the article indicates falsification of counter examples in all forms of notation of linear equations that are possible and significant in terms of teaching the subject Let s describe probably the most spectacular one A child participating in the process of mathematical education at school learns at its very beginning that such equations are solved in the following way We have to accept the situation in which the reply to student s question why should we do these particular transpositions is because it s been agreed so etc Let s do the following transposition in the equation ax b 0 x 0 We will get an equation rational notation of which seems to negate the theorem specified above On the other hand note that it is only formally different identity and not really different notation of the equation With this notation rightness of the theorem is evidently trivial Since rightness of a theorem is not determined by the form of the notation falsification of the counter example is proven According to the theorem the transposition of eliminating redundant factors is made in the order reverse to the rule of the sequence of arithmetic operations which means that redundant factors are eliminated by operations inverse to the operations in which such factors occur Redundant factors occur in involution and multiplication In multiplication it is number b therefore it is eliminated in the operation inverse to multiplication when both sides of the equation are divided by b In case of involution the negative exponent in the power of number x inverse of number x is redundant The inverse operation is raising both sides of the equation to the power of 1 Comparative review of the algorithm of finding solutions of linear equations Mathematisation of science and also scientific research and discoveries obliges scholars of all disciplines even those very distant from mathematics in their fields of scientific interest to use linear equations and formulas with one unknown freely due to linearity of a number of phenomena that are described by the science It is hard to expect that a social studies researcher could see and denote a linear dependence using an equation if he she cannot solve such equations independently Such skill that is training the algorithm presented in the article is also the caesura in the school mathematics education A school pupil relies in this matter on the intuition of the teacher who also solves equations by intuition as it has been described above The problems related with teaching mathematics beside the area of reporting facts are rooted more in the necessity of shaping intuition than in the intellectual limitations of a child referred to as the lack of mathematical talent XII XII A very important aspect of teaching mathematics at school is to redefine mathematical formalisms properly using a language that should be understandable for students which has already been postulated by A Einstein everything that can be simplified should be simplified but no more than it is necessary Author s note Before we get to the review of all forms of the equation ax b 0 significant in the school education let s solve the equation from the beginning of the article As it can be noticed it is equivalent of the following notation Eliminate the redundant factor number 1225 present in multiplication performing the division transposition Eliminate the redundant factor 1 exponent performing the transposition of raising both sides of the equation to the power of 1 Eliminate the redundant factor number 84 of the subtraction the transposition is adding 84 to both sides of the equation Eliminate the redundant factor number 15 of the quotient the transposition is multiplying both sides of the equation by 15 Eliminate the redundant factor number 7 of the multiplication the transposition is dividing both sides of the equation by 7 Eliminate the redundant factor number 30 of the subtraction the transposition is adding 30 to both sides of the equation Eliminate the redundant factor number 13 of the multiplication the transposition is dividing both sides of the equation by 13 Algorithm of solving linear equations Getting a significant result in elementary mathematics is commonly deemed to be impossible It so happens owing to the fact that mathematics is actually the only scientific discipline where a fact once proved remains an undeniable scientific truth until the end of its existence However it appears that even the queen of sciences contains certain oversights They are connected with the method of transferring knowledge which was consolidated by centuries of tradition and school routine What is most intriguing is the fact that the above stated truth concerns the most important mathematical research tool which since the beginning of mathematics 1 has been the linear equation Literature on the subject 2 3 provides only the tools one may use while solving a linear equation without explaining the sequence of steps necessary for the solution to be obtained One may only imagine why such an important problem has been omitted by mathematicians throughout three centuries of utilisation of the linear equation Since practically the lack of certainty of choice does not allow for an unambiguous determination of the procedure of solving equations what and when shall be done so that an algorithm is always successfully solved As a rule 2 3 the problem is explained as follows Linear equation is an equation where the unknown x appears only to a power with an exponent 0 or 1 and has the form ax b 0 Solution of a linear equation has the form x b a Solution of a linear equation can be found by a method of transposition in Polish literature a method of identity equations on the basis of the following theorems I Each linear equation can be converted identically into an equivalent equation II The same number can be added to both sides of an equation III The same number can be subtracted from both sides of an equation IV Both sides of an equation can be multiplied by the same number different from zero V Both sides of an equation can be divided by the same number different form zero The above presented theorems do not specify the minimal number of steps or sequence in which they should be performed so as to successfully solve an equation The lack of such a clear algorithm describing the steps required for finding a solution of a linear equation is truly one of the most surprising oversights in the history of science with an emphasis being placed on mathematics and its didactics In place of the algorithm proposed in this article mathematics textbooks contain only several examples of typical equations and their solutions after which they suggest proceeding to exercises If professional mathematicians and indeed mathematics teachers do not know the algorithm shown in this article and what they have at their disposal is an intuition without a name then one should not be surprised that they cannot explain the issue in an comprehensible way In didactics of mathematics at the elementary level it is a decisive moment for the acquisition of further matters and ability to

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