Abstact and concretization

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From specific to generalization and backwards. How understanding works. A research paper in Didactics of Mathematics.

Submitted: February 12, 2018

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Submitted: February 12, 2018





From the first steps of our education, we learn to represent objects of natural reality to notions of the mental world.We are taught a first representation.Specific corresponding to a meaning.The value of this representation lies as all things in its usefulness, which has to do with understanding.But the following question is posed.Is there truly a representation we learn before all the rest, even before the alphabet and addition?The answer may seem a bit vague, so we will examine some notions that will make things clear and will help the reader see the point of view of this text.

The learning of mathematics is seperated in phases.One of the most important is that in which the notion of number is introduced.This notion is not self evident, since the stimulus the human brain receives, come from specific objects of reality.The notion of numbers is a product of one of the abilities of human thinking.That of logical deduction (removal).

During this process, properties are withdrawn which ourselves have names in objects, based on our experience.We clarify here that for nature these properties (colour, shape, size) do not exist.Nature is not obliged to "think" like we do.The perception of reality in combination with logical deduction leads us, after a sequence of steps, to the listing of objects in sets or teams with common charecteristics.So if we speak about a forest and work with the above, we we can talk about coniferous or evergreen trees.While if we refer to a tree and with logical deducement thinking strip it of all its properties we reach to the number one.

Beginning with this point, we define the addition operation.But the procedure we followed is not the only way into the world of anstract.We could have begun through right from the start from the world of abstract and arbitrary define the notion of number and proceed to the operation of addition.If the definition of the number is correct and is verified (we can count, number and price since these are the everyday mathematics), then the operation of addition produces correct results if done by the rules of definition.For a mathematician this procedure is not something new.However, for a person who has not studied Mathematics, things are not so simple.Besides in the case of fantastic numbers, we prefer the abstract way and verification is not possible through experience.Only the results.We note that proportionately thase things stand in Geometry, when we have a directed line segmant and a vector.The choice between abstract and specific way of explaining, may be the difference between understanding and question.The specific way of explanation/transmission is expressed in the above by the example of the forest and the tree.This is the first representation even before the alphabet.The "transport" of abstract notions of the mental world in concrete examples of natural reality.

If we now theorize that logical deduction and concretization are the opposite procedures, one can see why the term representation was used.The goal in the pages below is to describe the representation in the context of mathematics teaching with the wider meaning (NOT equations) and point out its value stemming from the mechanisms of understanding when generalization and concretization take place.

Goals of teaching

Starting our approach, we can observe that the notions of abstract and example also, are credited in modern Mathematics as of fundemental value.Borrowing  some sentences from writer Peter Damerow we quote the following:

"It is generally considered a worthy goal for students to learn to think in an abstract way during learning process.It is only that practical experience imposes a number of restrictions: For one to be successful, it is necessary to start from specific examples and it is not advisable to go into abstract objects too early.All these sound completely innocent.However it is somewhat surprising that, even though the notions of abstract and specific are used constantly in the literature of mathematics education, the notions themselves are only rarely examined.As a rule, clear definitions are a goal.However a number of difficulties is obvious.

So for example, the regular didactics value that abstract thinking is a worthy goal is ambiguous.The purpose of teaching of mathematics is not only to encourage deduction and abstract way of thinking, but to apply the abstract notions and results to specific examples.So, "concetization" seems to be a meaning goal in the context of mathematics education, and one might add concrete way of thinking a worthy teaching goal."

Continuing his analysis, Peter Damerow presents definitions of the abstract notion, between them that which was used in the beginning of the text, widely known as Aristotelian.We will not research so much with definitions, as with the report that "concretization" seems to be a meaningful activity.Let us examine such a case.

The main volume of mathematical activity in everyday life comes down to measuring, calculating, numbering and pricing.All these using always the ten digit arithmetical system.So if we want to adress a class of students about an abstract notion such a binary arithmetical system, then we can assume with enough certainty that the usefulness, hence the value, of the above abstract notion is not known, since there are no indications that the students have from their experience any contact with the said abstract notion.Even though we can proceed in a detailed analysis of the rules of arithmetics of the binary system without giving any other explanation, this approch looks more dogmatic and authoritarion, since the notion is well away from reality and does not seem to be included in any context that justifies it, besides not being explained by the modern cognitive didactics of mathematics.So, no bad reaction will occur if we refer specifically and give an example about personal computers.Personal computers understand two conditions in which the information must be translated.When electricity passes through the circuits and when it does not pass.Corrolating or representing the first situation with the number 1 and the second with the number 0, we explain with a concrete applied example of the usefulness, hence value, of the binary arithmetical system, thereby convincing of value of learning it.This convincing "legalizes" the abstract notion in the real world, as the "what's point argument" does not hold anymore.In a similar way we can refer to the sixty number arithmetical system and the measure of time. 

Furthermore, following the sequence arithmetical system-binary system- PCs(personal computers)- arithmetical rules of the system, we succeed in guiding the learner's thought in the vague path: logical deduction-generalization-concetization-corrolation.So a simple second look, shows that the case we examined agrees with the words of Damerow.

At this point we must ask a necessary queston about the process of deduction and concretization.Is the role of the teacher necessary in this process?Can the student begin the thinking process on his or her own?

The answers come down to if professors of mathematics are needed in general.Besides the knowledge a teacher possesses can be found in a book or a software program.So why should the students be obliged to attend time-consuming classes during their education and spend lots of money to supplementary teaching?Would it not be easier and more economic to become members to a regural or electronic library and just show up for exams?

Many pupils read the lessons with little or no understanding.They see the words but do not understand the meanings.They need someone to tell them what is the important thing, give priorities to their ideas, demonstrate the techniques, answer questions.This is something which a book or a computer can never do.When a student asks a question, the professor listens and immidiately knows at what level he or she must give the answer.A look afterwards at the student's fave is usually enough for the professor to judge if the student has understood.A book or a PC can never interact with pupils this way.

So it begins to become clear, that in the context of mathematics and learning, deduction and concetization are unshakenly connected with the role of the teacher.But it would be an omission, if not "convenient" critique, if we would not question learning mathematics itself.It is a question that incurs immidiately if one remembers the binary arithmetical system example where we mentioned that the main volume of everyday mathematical activity is confined to measuring, numbering, calculating and pricing.

This kind of mathemaics is not but a fragment of mathematical knowledge.So for what reason do we learn so much other mathematics?Combining Damerow's views with personal, we will attempt an analysis.

The answer that mathematics are useful for sure does not hold in the level of everyday life, since after we learn to add, subtract, receive change,calculate, price goods and make the tax statement, there are not much else necessary for everyday life.A second answer that comes to mind is that mathematics are used in applications in science, industry and governing.The truth is that only a small percentage of people are going to need mathematics for physics,chemisry, engineering and other applications.So why should the majority of the pupils go through a process described by most as unpleasant for the sake of the few?There certainly must be more effective ways of choosing these people that will work in industry and commerce.If we want to be honest there are the of conclusions we reach.

There is of course the view that learning mathematics improves the mind.But in what way, it never becomes clear.But one could argue with certainty that traditional mathematics does not improve any other part of the brain, besides memory, since creativity, resourcefulness and researching are not results of normal mathematics teaching, either they are the old either the new mathematics,, since each is "served" in the same way.An authenticity-book dictates what mathematics exist for one to learn and an authenticity-teacher, or even a good and tolerating teacher, suggests how one should learn the mathematics that someone in an office of a ministry or a committee decided every child in the country must know. 

Which is the final result?We ask a person in the street what he or she knows or understands about mathematics and the majority of the answers is nothing or almost nothing.Besides a few noteworthy exceptions, most people hate or dislike mathematics and are simply impatient for the exams to finish so as to forget them.So if the purpose of teaching mathematics was the knowledge of mathematics, we could not have imagined a greatest failure.But if it is not the goal of mathematics knowledge, but something else, then what is it?Perhaps there are apsects of mathematics, which learning them would encourage the development of behaviours and specific ways of thinking.The evolution of some ways of thinking like deduction and concretization, as well as certain strategies that people can develop, would make them be and feel ready for situations they have not faced before.

A side of mathematics that clears the issue is the fact that mathematics have levels of generalization.Mathematicians are from their nature generalizers.Whenever a thereom is discussed, sooner or later the question is asked, under which conditions can the thereom be generalized to encompass wider sets of  numbers or objects.Such a kind of research has a great interest in a rapidly evolving world.

When we have a number of situations, we need to know up to what extend can we keep the properties of objects or events that take place while encompassing a wider set of objects.In the Internet era of social networks, this is translated like the difference of Facebook and Twitter.In Facebook your messages, hence socializing, is limited(mainly for security) to friends.In Twitter anyone can view your messages(but you know that from before), so you end up socializing with people from very distant places.So the bottom line of generalizing is that you get bigger sets of objects or people in which the same properties hold.In other words, we are allowed to create a greater class of "facts" in which we keep our advantages, in a way so as not lose "profits".This is done very often in mathematics.So the moving from specific to general and backwards again, is an important an interesting capability that we would expect to be the product of learning mathematics.Crossing Damerow's views, this would include not just "serving" the students generalized situations, but allowing him or her to generalize.Like he or she learns to subtract by subtracting, so can he or she be taught to generalize by generalizing.

It must be sublined that deduction(removal) and generalization, even though they are seperate qualities of human intellect, their development usually happens similtaneously but indepedent of each other.The kind of practical questions that would incur from this sort of instruction, would be "what would happen if" or "assuming that".Real problems do not have in practice clearly correct answers even though they have wrong answers.This kind of "what if analysis" is definately flexible and useful in administration,industry and scientific research.Modern man is forced to examine the possibility of change.The truth is he has no choice but to do so.The altenative case lies opposite of adaption and is the stasis, when we live in an ever-changing world.

But during adaption, wondering what would happen this or that condition is not a question of if.In reality the conditions will change.So the matter comes down to what is likely to happen.Reaching a conclusion, if mathematics is possible to constitute an education of "what if" way of thinking, then this would a noteworthy reason for the teaching of mathematics to children, as opposed to traditional shallow justifications of mind sharpening and application, which in the end play a smaller role.

The approach of knowing the value of learning mathematics is very important in the opinion of the author, since besides that convincing is the essence of democracy, how can one learn something if he or she is not persuaded why?

So summarizing the above, deduction and generalization are fertile expressions of the mathematical way of thinking, with the teacher being the guide developer in a "here is the value" specific learning process.

 Learning mathematics

Learning constitutes the most fundemental goal of teaching.In order to succeed, we must scout the interaction mechanisms of deduction and concretization, so we clearly define how understanding and assimilating happens concerning these two notions.Else we are in danger of turning instruction to unproductive areas such as neverending discussions and pointless exercizing, while degrading thereotic tools like examples and proofs.Learning is not something self-evident and understanding does not come only from reading a book.Problems as well as phenomenona(which deceive) arise in learning mathematics, which imply the mechanisms of assimilation are well hidden.The problems may be of pathological nature(dyslexia) and pedagogical nature(literacy), fields we will not get into.Or of mathematical nature(circular thoughts, thereoms accumulation).As for the phenomena, isn't it strange how some students are considered mathematically gifted, while an impressive percentage faces serious difficulties?Because if the gifted are the exception and the rule are the weak pupils, how is that we are doing everything right? 

A field in which we will try to find answers is mathematical psychology.If learning is defined as the process of acquiring knowledge, then the appearance of the first models in 1907 according to Gulliksen(1934) explains some things.The first model was the predeccesor of the kind of model based on the assumption that acquiring knowledge happens in a constant rate but oblivion is proportionate of the existent knowledge.These hypotheses imply that the learning rate is proportionate to the knowledge remained to be learned.Specifically, the learning curve(the costant curve of errors as a function of the number of trials) will decelerate.Which simply means that we refer to learning as a rate of information(plus is learning, minus is forgeting).The fact that most curves of learning based in data concentration are decelerating(negative rate of information) was suggested in support of this model.To put it simply, most learning curves say we forget details and dates but remember facts and ways.So learning must be a pool of information and data.

Two observations must be made.First, the problem of learning mathematics as accumulation of thereoms and memorization of proofs is explained well with the above model.Second, remarks that the mathematician Dienes has made concerning learning traditional or modern mathematics, which say no result comes out of it, seem not to lack basis.If knowledge of mathematics is acquired through memorization, then the message coming from the "learning as a pool of information" model, is that this way of appoaching instruction is doomed to poor results.Something that is confirmed from the "amnesia" of pupils and students right after the exams.The above model is also found and in areas outside psychology.For example, it can describe the balance account of a person with steady income who spends in a rate propotionate to the balance amount.

An alternative model of late 20th century, assumed that learning took place with a rate proportionate to the knowledge already acquired, but oblivion was proportionate to the square of the knowledge acquired.In a few words, learning becomes harder as time passes and more and more data are stored in your memory.You tend to forget details of your youth years and technical information.This model is also found in areas outside psychology.It can describe the spread of a epidemic, as more people exist to transmit the disease, but which decelerates as the number of healthy people diminishes.The process is self-catalyzing, as it activates itself and consumes its own resources. 

Summarizing, it is ephasized that these models have not contributed greatly in the evolution of learning theory, beyond an important feature of learning, the learning curve.In short, how learning is decribed with time.However, as a comment it must be underlined that they give an extra dimension to commonly accepted remarks and generally accepted problems.You cannot solve that you which do not understand.

A second approach in deduction, generalization and concretization constitute the mental aspects of the above procecces and give some interesting information.Dienes defines generalizing as "the discovery that a general rule is extended beyond the few first cases", while he states for deduction "it is the consience that the rule is applied to a number of other situations".He attempts to order the various levels of generalizing with the following way:

1.The extention of the rule from a finite number of cases to an infinite number.

2.The generalization from one infinite class to an another infinite class.

3.That which Dienes puts as "the mathematical generalization or the forming of an isomorphism(for non-mathematecians, different oblects but same structural properties) between a class and the subclass of another class.

To clarify things, we present the exaple of function


If for each value of x , the value of y can be found, then after a finite number of repetitions, an extension can be made that for any value of x we can found the value of y.That is the first phase.The second phase is achieved when the factor of x is changed to any number, which means it belongs in an infinite set.The first rule of course must certainly be maintained.So we write:

(A)y=ax ,a belongs to { 1 , 2 , 3 , ..... }

The third phase corresponding to the third case of Dienes generalization cases is the generalizing:

(B)[y]=[a][x] ,where [y], [a] and [x]  1x1 matrix

Here the first case (A) is a subclass and isomorphism to the second case (B).

Dienes claims that a mathematician experiences s feeling of strength when he or she understands a generalizing.The extension of a rule to an infinite number of cases is followed by a sense of wonder and an energy release.The successful attempt in generalizing, gives the pupil the ability to write a sentence that includes a very big amount of information.This success is the source of the strength.

Without making an error(by chance) effect, the achievement of generalizing may release storage space in the central nerve system, since generalizing can assumed to be a more effective encoding of information.A big amount of information takes corresponsively a big amount of space.But at the amount of success(when understanding actually happens), storage space is liberated, releaving the pressure on the nerve system, creating a fullfilment joy feeling, that's why the "Oh, now I get it" reaction.The central nerve system is not a hard disk drive.If there is empty storage space, it could rearrange information, so as to use all space and reduce the pressure to the rest nerve system.Besides creating this way a new structure of knowledge as Piaget claims knowledge and learning is constituted, the climax of "action-reaction" and relief of pressure  may explain the internal mechanisms of motive during learning mathematics as well as why to some children like mathematics so much.

From this point of view, deduction has an important role.If it is assumed as the rejection process of information and "noise"(in terms of theory of information), then this leads to the real message of encoding.The latter may be immidiately perceived intuitionately, but the truth is that it is clear deduction crosses generalizing perhaps in a right angle.At this point, we will make an analysis of the structural theory of education and the view of Piaget.

Cognitive psychology transfers the weight of teaching, from the reaction of the pupil to relative actions in the organization of instruction.To his or her capacity to interact with the subject being studied and the social enviroment(the rest of the class) during learning.The pupil is not adressed as directly dependent of the enviroment.The enviroment is affected from subjective interpretations and these guide actions.Each piece of information that is acquired and consequent action, assumes the assimilation of data in cognitive structures.These structures do not simply exist in the student's mind.They are based in systems of generalized rules that allow the structures to activate when applied to seperate and specific cases and are concretized to the necessary degree.In this sense, each system of generalized rules of interpretation represents a closed system of possible relations of derived from experience, content.By assimilating to a system like this, the information -the isolated fact- is introduced to a network of dependent possible actions.

There are two apects included to each learning process that must be seperated.The assimilation of new experiences and the adaptation of the structures themselves.The adaptation is not a continuous and automated process.It is interrupted by the observed phases of learning, where understanding takes place and the structures are reorganized.This restructuring is connected  to the letter and/or spirit with reinterpretating areas of experience.The development of the phases of reorganizing of the cognitive structures is decisive for the long term success of educational process.

In the above context, assimilation of new experience can be made with different ways, which will be examined.As a starting point, we imagine that understanding is not a reaction to a behaviour request(like the expression of anger or a normal conversation).But an active achievement of the student(he or she questions, explains how he or she understands, requests confirmation, verifies).So on case-by- case basis where a new notion or solution of a problem is assimilated to the already existing cognitive structures of the pupil, things are simple.Either the former experience of the pupil is concretized and formalized to the new notion whereas the pupil fuctions productively, or with specific criteria the pupil judges for himself or herself, if the interpretation of the notion has assimilated to structured system of the already existing knowledge, whereas the pupil understands after reasoned critique and thinking.

If the notion is too foreign or revolutionary to the existing cognitive structures, the reorganizing of the structure, hence the learning, takes place.The latter sentence is the essence of Piaget instruction and learning theory.

Because understanding and learning happens when things get connected, if one connects Piaget claim with Dienes argument of better encoding and nerve pressure relief, it becomes visible how things work to certain degree.But this a correct but incomplete instruction theory.The author argues that a specific practice is needed.Which comes in the form of fertile stimulus.The teacher gives the fertile stimulus but what the pupils undestands is its own matter.However, knowledge is not vague but specific.So acting in the way a teacher is needed as we pointed out above to prove and explain the teacher's value(argaize ideas,give priorities,demonstrate techniques,answer questions), the professor guides the pupil into understanding the knowledge.This is author's and paper's educational argument.This approach is based on freedom and its main achievement is that it develops judgement in the pupil's and not the teacher's criteria.Hence, it is an instructyral approach aimed at learning revolutionary and different kind of class notions.The notions that a reorganization of the cognitive structure will take place and this "free judgement" instruction approach is needed.Like the negative numbers taught to high school pupils, after they have learned the integers and the fractional numbers.The fractional numbers taught by the pizza example is a poor way, as it creates a "paradox" effect.It does not make sense or can be explained how it connects to the negative numbers taught by the debt example(also poor way).The left side of number zero of the " straight horizontal real numbers line" is a better way(integers are points on the right side and fractions like 1/2, the middle point, so afterwards for negatives, you simply talk about the left side) as negative numbers are a fundementally new notion.As such, it is always hard for pupils to understand, and the disillusionment with mathematics begins, developing to  "amnesia" right after exams.

Dealing with the latter problems was the reason this paper was written.You cannot stop what you do not understand and in the same way you cannot improve a situation until you decide the goal, know how things work and devise a way to do it.

Book references

  • « Abstract and Representation », Peter Damerow , Kluwer Academic Publishers
  • « Mathematical Education » ,Van Nostrand Reinhold 1978 , Z.P.Dienes , edited by G.T.Wain
  • « Mathematical Psychology , An elementary introduction » , Amos Tversky - Clude H.Coombs -  Robyn M.Dawes , 1970
  • « The process of learning mathematics » , edited by L.R.Chapman , C.Plumpton , Pergamon Press 1972
  • « Perspectives on mathematics education » , Reidel Publishing Company 1985 , W.DORFLER  and R.R.McLONE
  • « How to teach mathematics : A personal perspective. » , American Mathematical Society 1993, Steven G.Krantz
  • « Calculus I » , E.Pouleas, Thessaloniki 1998

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