In English reference books - transposition, in Polish publications - identity operation
, is a nontrivial form of the equation ax + b = 0
(in contrast to its trivial form, e.g. 2x + 3 = 0). Its solution is x = 45. Solving should start with dividing both sides of the equation by 1225 (why?), while the readers will probably come up with other procedures leading to a successful solution. We insist on solving this equation because through this process the readers will realise the challenge of the children and youth mathematics education which is explaining the issue: why equations are transposed this way and not the other in order to get the solution. Solving this particular problem requires using only the most elementary tools of school arithmetic: subtraction, multiplication, division, fraction, brackets, inverse of a number!
Twenty years ago, the equation
was one of the five typical tasks during secondary school admission tests in Poland - to be solved in approximately 15 minutes without using a calculator. Estimations have been made on the basis of author’s individual research and the statistical analysis of the results of the exams for the certificate of secondary education prepared by the Central Examination Board of the Ministry of Education of Poland, published on their official website.
I. Scientific truths: creativity or discovery
For centuries scientists (IV) have been absorbed in finding the answer to the question:
“Does mathematics (along with scientific truths) exist and is only discovered, or it is only a product of original creativity of individual scholars?”
(IV) Reference books, for instance (1) usually begin with Pythagoras ("the world is a number") and Plato - as the forerunners of this way of thinking about mathematics. However, it should be taken into account that there are no sources concerning the ancient times that would be credible in terms of the methodology of scientific research. A Polish researcher, M. Kordos in (2) justifies a more critical approach to the hypotheses concerning the ancient times in the following way: “[...] Apparently the first director of this library (refers to the Library of Alexandria, author's note) was Euclid. It is astonishing how little we know about him. It is claimed that he came from Megara (as a result of confusing him with another Euclid) and lived between the years 365 and 300 B.C. (most frequently given years), it is believed that he studied at the Platonic Academy (just as every self-respecting scholar of those times...) and that he was very well educated within the scope of the (already) ancient Egyptian knowledge (since Alexandria is located in Egypt), etc. However, all attempts to confirm any of these theories using at least two sources have been unsuccessful.” Author's note.
A consequence of choosing only one approach to practicing mathematics and also science in general may be
heuristic inertia(V):inability to make certain fundamental discoveries.
(V) Heuristics has become significant thanks to George Polya, eminent American mathematician, who claimed in (3) that the ability to make scientific discoveries may be taught, beginning even in the early childhood, at least in case of mathematics. Author's note
For instance, in the scientific report cited below it refers to
the algorithm theorem(VI), finding a solution of a linear equation - an issue which is fundamental for science and education(VII)
(VI, VII) The term of algorithm in the article refers to formulating a list of steps that are necessary and sufficient to get a solution of an equation with one unknown, justified by the scientific methodology.
It is astonishing that the ability to use one of the most significant tools of not only mathematics but the science in general, that is the equation with one unknown, must be taught by intuition during the school education, because there is no such algorithm. It is relatively time-consuming and takes many years. For example, in Poland 6 years old pupils get acquainted with linear equations finding solutions to simple equations, such as: 1+ ? = 3, After 12 years of further education, they end up with solving linear equations noted in a rational form: a = - b/x ! Intuitive teaching of school mathematical competences results in removing them from memory very quickly. R. Feyneman commented on this issue: “There is something wrong with the people: they don’t learn by understanding but some other way, perhaps using their memory. Their knowledge is so weak”. The research carried out in Poland by the author has indicated that after 20 years most secondary school graduates are able to give an algorithm of solving quadratic equations in a trivial systematic form: ax2+bx+c = 0 („[…] Δ=b2-4ac and x1 i x2”)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 ax2+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: axn+bxn-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:
- 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 find solutions not only of mathematical problems, but also those concerning physics and chemistry: “get it or die.” The lack of the following theorem: “In order to find a solution of a linear equation, we use the method of transposition, getting rid of unnecessary factors in an order reverse to the order of the operations we perform”,
and the lack of a systematic overview of means by which one can find solutions of all types of linear equations are some of the fundamental reasons behind the low level of mathematical education, and the fact that mathematics itself is deemed as an extremely difficult subject. It shall suffice for a student familiar with the aforementioned theorem to remember and adhere to the two-step-procedure presented below in order to always be successful. The first step of the procedure requires an overview of arithmetical operations present in the equation in order to determine the primary operation, i.e. the last one to be performed in accordance with the order of arithmetical operations. The second step consists of performing an operation on both sides of the equation opposite to the primary operation in order to get rid of this part of the primary operation where the unknown does not occur. Repetition of the two-step-procedure always guarantees success with the use of the smallest number of transpositions.
Let us present the procedure with the following example:
We repeat the procedure:
|ax + b = c , x=?||1. We have here two operations: multiplication and addition. The last operation to be performed (i.e. +) is known as the primary operation.
|ax + b = c | -b||2. Now, we get rid of this part of the primary operation where the unknown does not occur, performing an operation on both sides of the equation opposite to the primary operation, i.e. we subtract b from both sides of the equation
|ax = c – b||
|ax = c – b||1. We identify the primary operation: multiplication (•)
|ax = c – b |:a||2. Now, we get rid of this part of the primary operation where the unknown does not occur, performing on both sides of the equation an operation opposite to the primary operation, i.e. we divide both sides of the equation by a
As it is easy to notice, in order to use this procedure, it is enough to know the aforementioned theorem and be familiar with the order of performing the arithmetical operations. The accessibility of this procedure renders the whole issue easy and intuitively logical both during the acquisition and explanation process. The proof of this theorem is obvious, and it reverses the traditional order of approaching an equation from its form to its solution. We assume that the solution of the equation is primal, and that its form is a consequence of arithmetical operations performed on both sides of the solution. While building a material conceptual model for students, we can present the form of an equation as its coded solution which was created in the following manner:
- We assume that an existing solution x has a concrete numerical value assigned to itself.
- We postulate that an arbitrary form of a linear equation shall be obtained e.g.:
- We determine the order in which operations will be performed on a postulated structure: multiplication by number a and addition of number b, and we apply the determined order to x = k:
|x = k | • a||x = 2 | • 6
|ax = ak | + b||6x = 2 • 6 | + 1
|ax + b = ak + b = c||6x + 1 = 2 • 6 + 1= 13
- reversing the procedure we obtain x = k and x = 2, at the same time proving the procedure of finding a solution to the linear equations in a manner accessible for students.
Let us provide here an overview of all types of linear equations:
Presenting these 20 types of equations to the students in such a hierarchical order allows the teacher to stop the presentation at any given moment of the educational process and practice chosen equations until students become proficient at their solution and reach the level of comprehension. At the same time, such a presentation makes it possible to avoid the axiomatic method – the nightmare of every mathematics teacher: “because that’s how mathematics works,” allowing for an accessible explanation of “why it works the way it works.” Obviously, the procedure in question was known in case of these twenty equations; however, taking into account the lack of the theorem in question, as well as several special cases discussed below, the authors of mathematics textbooks have so far failed to present the procedure as effective also in solving those remaining cases. It simply requires an appropriate analysis. The didactics of equation solving in case of mathematics taught at school is spread over many years (the basic teaching program of mathematics in Poland spreads the issue of the linear equation solving over a period of 12 years) – at various levels of education, various authors discuss only those parts of the issue which are appropriate according to the teaching program of a given textbook. It, in turn, results in fear on the part of students and lowers their self-esteem which leads them to a conclusion that the issue of the linear equation is difficult and so is mathematics as a whole. As has been shown on the basis of the aforementioned examples, such a presentation of the issue allows even the students with poor skills and competence in the filed of mathematics to understand the procedure behind linear equations solving and it does so at the very initial level of mathematical education. Below, we are going to present four types of linear equations which only require analysis sufficient for the application of the two-step-procedure:
At this point, students usually come up with a question: why do we subtract a from both sides of the equation? It is enough to show that a – bx = c is an abbreviated notation (a transposition) of the equation: 0 + a – bx = c. One may also handle the problem from the point of view of didactics, introducing the rule of equation arranging before proceeding to solve the equation. Then, the rule can take the following form: we arrange the polynomials starting with the highest power of the unknown and finishing with the lowest one. In the case of linear equations, we will require such an arrangement of the equation where the unknown is positioned at the beginning of the equation. The postulated procedure is valid also in these cases.
22. ax + bx = c
The previous examples of forms of the linear equation notations had the unknown x appearing only once in the equation. At this point, it is enough to remind the students of a certain intuition:
5 = (2 + 3) = (1 + 4) = (7 – 2) etc. No matter what objects we compare by means of addition and subtraction (apples, stamps, money, x), we still have:
2apples + 3 apples = (2 + 3)apples 2x + 3x = (2 + 3)x
Once we are sure that a student can use this tool freely and competently, it shall be easy for him/her to understand the following transposition:
ax + bx = c <=> (a + b)x = c
and use the easily accessible procedure of two steps explained above – always leading to a success.
23. ax – c = bx + d
It is the third form of linear equation which causes difficulties in acquisition for the students, thus making comprehension of the solution of the linear equation even more difficult: not only does the unknown occur in the equation more than once, but it is also present on both sides of the equation. Presenting, as was the case with example 22, several transpositions of the following kind:
x = 3 |+x
x + x = x +3 <=> 2x = x + 3
2x = x + 3
allows the student to arrange the linear equation in such a manner as to be able to use the explained procedure of two steps, and facilitate practising and understanding of the linear equation solving.
24. The last form of the notation of linear equation which causes difficulties both to the didactics of this issue and in school practice, i.e. in relations between the teacher and the student, is the rational form (fractional form) of the notation of the linear equation which in the simplest manner can be presented as follows:
Because of the hierarchically unorganised manner of introduction to the didactics of the problem, students have difficulties with memorising and understanding why textbooks suggest the following operations:
The problem can be easily overcome by a reminder of an intuition according to which:
Reminding the student of this intuition will allow him/her to notice that it is not even the transposition of the initial equation, but simply a different type of its notation, which let us apply the aforementioned procedure of two steps:
It additionally facilitates transformation of physics formulas which happen to have a form of such a linear equation:
Many experts (1) for decades have been postulating a need for radical changes in didactics of mathematics which is rooted in patterns crystallised by school tradition and a way of writing mathematics textbooks established two hundred years ago, and which enjoyed their greatest successes in a society ruled by a completely different laws from those currently effective. The effect is that textbooks define hundreds of concepts in an entirely incomprehensive, and sometimes even illogical, way. Experts from various scientific disciplines do not seem to mind it, e.g. in Microsoft software, in order to switch off the computer, one has to direct the cursor at the start icon – only somebody who performs this operation for the first time is surprised. This case is similar to the teaching of mathematics. Those currently taught at school have to be simply written again from scratch.