THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS: A SOCRATIC EDUCATIVE EXPERIENCE

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS: A SOCRATIC
EDUCATIVE EXPERIENCE
María Ángeles Navarro
University of Sevilla, Spain
manavarro@us.es
Pedro Pérez Carreras
Polytechnic University of Valencia, Spain
SUMMARY: We present a semi-structured clinic interview designed to ease the mental construction of a suitable concept-image of the notion of convergence for series of positive numbers. Cognitive obstacles will manifest themselves along the interview and we shall deal with them and teach the student how to overcome them. A special computer generated tool has been designed to be used by the interviewee to help provide plausible answers to the questions posed. Several situations will arise showing the limits of our colloquial approach and visualizations provided by the computer tool and we shall discuss why a mathematical theory is necessary if what we seek is to provide exact answers with certitude.
Key Words: accumulation, conjecture, series MSC 97C30, 97U50
1 Objective
Our purpose is to introduce the student to the topic of convergence via a Socratic dialogue. We want to explore how far progress can be made in providing meaningful information by using colloquial language while manipulating a computer generated visualization tool. Although it is unclear how those manipulative aids affect cognitive functioning, its use somehow provokes less disappointment if the machine proves them
wrong, favours the appearance of conjectures and frees students‟ thought processes. This approach will prove encouraging but will show its limitations too, mainly when asked to leave the safety of conjectural reasoning and jump to exact reasoning. The cognitive structure associated to a mathematical concept which includes all mental images, visual representations, experiences and impressions as well as associated properties and processes is called concept-image, (Tall & Vinner, 1981). Learning, understanding, applying and developing mathematical concepts involves the construction of this kind of structure in the mind (Noddings, 1990; Piaget, 1977). By means of a semistructured clinical interview, we shall provide the means for the construction of a solid concept-image of convergence of a series of positive numbers which incorporates visual, numerical and algebraic connotations. For this purpose and with the help of a mathematical assistant, we have designed a tool covering all those aspects. Our aim is not to develop a
substitute for the concept-definition (the conventional linguistic statement precisely delimiting the frontiers of application of the concept, Tall & Vinner, 1981), but rather a somewhat narrower objective: to describe a battery of actions that should be implemented prior to formal mathematical instruction in the classroom with the purpose of, on one hand, constructing a suitable concept-image which does not distort the desired concept-definition allowing an easy transition to it and, on the other, provoke the need for such a conceptdefinition. The natural places to implement such a strategy are late High School or first year of College. It is not our purpose to measure the extent of the student‟s previous knowledge on this specific subject, but to guide them in a journey of discovery testing at every significant stage of the experience the foundations of their beliefs or their ways of reasoning, very much in the Socratic spirit.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
2 Convergence
Many articles are devoted to the task of understanding limits. Learning the concept of limit informally, students can be helped to construct images about the infinite process of a sequence itself (Navarro & Pérez Carreras, 2006). Examination of primary intuitions related to the notion of limit of a numerical sequence in secondary school pupils who have not yet been exposed to its more formal definition can be found in Sierpinska (1985 and 1987) and Navarro & Pérez Carreras (2006) where considerable divergences between convictions and concept-definition are pointed out. An introduction to the idea of convergence via decimal expansions can be seen in Navarro & Pérez Carreras (2010) showing conflicts arising from sequences seen as indefinite and ongoing processes without considering them as the result of such processes. In Robert (1982 and 1983) several models of convergence of numerical sequences in university students are presented. Partially similar observations were made in other writings dealing with understanding the limit of a sequence by pupils and students (Davis & Vinner, 1986; Tall & Schwarzenberger, 1978).
Students‟ conceptions related to the notion of limit of a function can be seen in Cornu (1981 and 1991), Ervynck (1981), Ferrini-Mundy & Graham (1994) and Williams (1991). Students‟ images of limit are pointed out as having influence on understanding the ε–N definition of the limit of a sequence and the complexity of its logical structure makes it difficult to understand why the quantifiers should be described in such a way, (Cornu, 1991; Mamona-Downs, 2001). Even if this definition has been mastered, students tend to continue to use images of limit that are associated with their previous learning experiences,
(Pinto & Tall, 2002; Przenioslo, 2004).
3 Framework
We shall anchor our experience in Van Hiele‟s educational model (Van Hiele, 1986 and 1959) which provides a description of the learning process, by postulating the existence of levels of reasoning (not identified with computational skills) classified as Level 1 (Visual Recognition), Level 2 (Analysis), Level 3 (Classification and Relation) and Level 4 (Formal Deduction). In Level 1 students are guided by a series of visual characteristics and lead by their intuition. In Level 2, individuals notice the existence of a network of relationships. This is the first level of reasoning that can be called “mathematical” because students are able to describe and generalize through observation and manipulation properties that they still do not know. Reasoning in Level 3 is related to the structure of the second level and conclusions are no longer based on the existence or non-existence of links in the network of relationships of the second level, but rather on existing connections between those links.
Level 4 speaks for itself.
Information about this model and its application to the study of geometry can be seen in Burger & Shaughnessy (1986), De Villiers (1887), Fuys et all (1988), Mayberry, (1983), Orton (1979), Orton (1995), Presmeg (1991), and Senk (1989). Its success in dealing with concepts outside of the realm of geometry (see Campillo & Pérez Carreras, 1998; de la Torre & Pérez Carreras, 2000 and 2001; Jaramillo & Pérez Carreras, 2001; Llorens & Pérez Carreras, 1997; Navarro & Pérez Carreras, 2006) is explained because the model is more concerned on how students think about a specific topic than with the topic
itself and also because the model mimics the genesis of some mathematical concepts: first, the discovery of isolated phenomena; second, the acknowledgement of certain characteristics common to all of them; third, the search for new objects, their study and classification and, fourth, through consideration of examples and counterexamples to proposed definitions, the emergence of definitive formulations.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
The application of this model to a specific subject requires the establishment of a series of descriptors for each level to enable their detection. To be considered within van Hiele‟s model: (1) levels must be hierarchical, recursive, and sequential (2) levels must be formulated so that they include a progression in the level of reasoning as a result of a gradual process, resulting from learning experiences (3) tests designed for the detection of levels should take into account the existing relationships among levels and the language
used by apprentices and (4) the fundamental objective of the design must be the detection
of levels of reasoning, without confusing them with levels of computational skill or previous knowledge.
4 Prior to the interview
According to the constructivist perspective which focuses in individual thinking, all that is relevant in the model can be accomplished via an appropriate Socratic interview design that, within the context of the model and paying attention to many of the recurrent themes described in the research literature, allows the detection of students‟ levels of thinking with respect to the specific mathematical concept we are dealing with. To carry out our study we elaborated a questionnaire for a clinical semi structured interview, allowing for our intervention depending on students‟ answers and providing us the opportunity to turn the interview into a learning experience. The most challenging aspect of the interview‟s design was how to be prepared to fight the attitude of most students which are comfortable with inconsistencies, contradictions and competing meanings which were the main cause of failure in completing the interview as our questions grew in sophistication. To favor the student‟s own construction of insight, we conceived it to resemble a journey which can be described as the plot of a good epic: be confronted by a challenge, leave the safety of your narrow previous knowledge, bond with new ideas, survive showdown and discover true self. Starting from a naïve understanding of the concept, we searched for a transition to higher forms of reasoning by placing it in different contexts (from verbal to visual, from visual to computational) with the purpose of facilitating, through abstraction, the construction of a definition with the help of a computer-generated tool. Our previous work in Navarro & Pérez Carreras (2006) was instrumental in designing a tool exhibiting two screens (one visual, one computational) which manipulated properly should help the student in providing answers to our questions as well as offering a visualization of the problems treated. The tool (a graphic user
interface) is an interactive screen and the user does not need to have any previous knowledge on the program (MATLAB 6.1) used to design it. The screen shows two windows, a graphical and a computational one: the first allows the plotting of partial sums as staircases and the other shows the numerical value of those accumulations. Two editable text boxes allow the typing of analytic expression for the series to be considered and the number of steps to be seen. Several buttons draw barriers to their growth as horizontal lines and allow changing scales and zoom. Several partial sums can be plotted and calculated simultaneously. Around twenty interviews to students in their first year at the University were carried out. They agreed to the audio recording of the interviews and also to the use of their corresponding transcriptions in our analysis. Their previous formation in Mathematics was the habitual one of students completing the Baccalaureate in Spain which nowadays does not guarantee any special solvency, unfortunately.
5 Anatomy of the interview: an overview
Let us describe the structure and goals of our interview step by step.
1. We start by showing several processes in mathematics which can be repeated as
many times as desired. For instance, taking a segment of unit length and proceeding with a

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
ruler and compass, we divide the segment in two equal parts; we consider the right half and proceed to divide it again and so on (Figure 1). Figure 1: Division process of a segment of unit length. Measuring the lengths obtained, we ask for a suitable symbolism representing the growing length provided by the accumulation of subdivisions. The figure itself suggests the expression 1/2+1/4+1/8+… and we ask which meaning can be assigned to the tail of dots and if he can predict the next term in this process and, for that matter, any subsequent one. 2. We may think of a progressive accumulation of quantities regardless of the
existence or not of a geometric setting suggesting why such accumulation takes place and we shall call it series. Each quantity to be added is called term and, since we are unable to write all terms to be accumulated, we may write just three terms and a tail of dots, only if we are able to predict the following ones. We point out those expressions such as 1/2+1/3+1/4+… are also series, even if they do not arise necessarily from any pictorial procedure. 3. We announce that our purpose is to determine whether a series produces a number in some sense, i.e. if the procedure of adding quantities in a progressive manner has, what we may call, an end product, that is, a number encapsulating all additions. Concerning our first series arising from a geometrical procedure, we ask if an equality such as 1/2+1/4+1/8+… = 1 makes any sense and, if affirmative, which sense would it be. We want to know how is this equality understood and whether there is any ambiguity concerning the equality in the algebraic sense. 4. Accepting for the time being the reasons why the equality is true in some sense, we proceed with some bi-dimensional figures (substituting lengths for areas) asking to provide conjectures of which numbers can be associated to the series appearing as the result of the indefinite accumulation of areas (Figures 2 and 3). Figure 2: Visualization of 1+1/2+1/4+… Figure 3: Visualization of 1/4+1/16+1/64+...

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
5. We engage in a discussion on which mental process lead to his conjectures: the acceptance of a mental process of adding quantities indefinitely, even if physically such possibility is rejected, and the guessing of the “total” sum, based on the depletion of physical space in the figures left by our step-by-step covering of terrain. Translating this procedure to a pure numerical problem, what we have is a step-by-step accumulation (addition) of quantities which seem to approach numerically a predetermined number
(length of the segment, area of the square) suggested by the companion figures. We ask how descriptions such as “after an indefinite accumulation nothing is left behind” can be reformulated as „the predetermined number being larger than any accumulation but also the smallest of all numbers with this property‟ or “the predetermined number, being not one of the accumulations, is the closest to all of them”. We point out
that, to be algebraically sound, we need to precise what “closest” means, which runs in two stages: First, we shall request him to explain what does it mean that a number B is larger than another number A, arriving to “there is a backward movement from B that stays larger than A” and we will request him next to define, taking into account this formulation, what means that B and A are equal trying to obtain a verbalization such as “any backward movement from B stays smaller than A”. Second, we shall look for an explanation such as the predetermined number satisfies that, for any backward movement from it, there is an accumulation larger than the number produced by the backward movement”. 6. Our next goal is to consider series which do not arise necessarily from any geometrical situation. We ask him to deal with an abstract one and what kind of symbol would be appropriate to denote it, ending with an expression such as a1+a2+a3+…, pointing out the role played by the sub-indices. Prior to discuss what kind of questions we want to ask regarding a series and to avoid future misunderstandings, we engage in a clarification of nomenclature and we ask him to elaborate on the distinctions between several similar looking expressions:
(a) an as the nth term of the series where each and every one should be known.
(b) a1+a2+a3+… as a progressive accumulation of quantities a1, a1+a2, a1+a2+a3, …
(c) a1+a2+a3+…+an as the nth accumulation.
(d) a1+a2+a3+… = x as an equality where x stands for the closest number to all accumulations (but this time, without a clue about which number is x going to be). 7. Having in mind our previous considerations on how to proceed to decide whether a chosen x makes the equality valid in a1+a2+a3+… = x, we are in desperate need to have a conjecture for x on which to check upon. To obtain a suitable candidate, we introduce him to the following visualization of our situation (Figure 4): Figure 4: Visualization of a series with our interactive screen.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
A few words about our prediction device are in order: shortly, there are two screens, a visual one showing the progress of the accumulations and a computational one showing the ordered numerical values of the accumulations from top to bottom. Due to the limited number of digits available, we warn him that both screens are showing only approximate values for the accumulations, but reasonable good ones due to the number of digits allowed. More detailed, once a particular series an= is introduced (for which an editable text box is available), the student can add as many terms of the series as desired, hence drawing and calculating as many accumulations as wanted. In our visual design, the screen shows the accumulations as heights positioned from right to left (each term an is added at the abscissa 1/(n+1)) joined by horizontal lines producing a staircase where only the heights of the steps are significant and not the widths of the steps. He may want to see only some portion of the staircase, for which purpose a pair of small editable text boxes are available. The number x which we seek is, if existent, the height of the point in the left vertical line to where the staircase is aiming at and therefore a zooming capacity is available by pressing a button to refine possible conjectures. A change of scale is also available. Other buttons allow the student the drawing of horizontal lines at the desired height as well as bands downwards from it. If a certain line is chosen as a candidate for x and if a certain band is drawn, a message in the upper board will inform him on the number of steps of the staircase lying outside the band (Figure 5). Figure 5: Using our interactive screen. Now we are prepared for some serious work. 8. Turning to our previous examples, we encourage the student to use our tool to confirm what he already knows as well as predict end products for other series. 9. Considerations on the “size” of the terms of the series are in order. By checking the visual screen with the series 1+1+1+… it will be clear that there is no hope of making the equality 1+1+1+… = x valid for any value of x. For any series a1+a2+a3+… we expect the student to state that the existence of x is equivalent to the existence of a barrier the staircase cannot cross. 9.1 In order to achieve that, some degree of “smallness” has to be present in our series from a certain position onwards. We ask the student if terms being small from a position onwards are enough to ensure the existence of a barrier. Once clarified that this
is not the case, the student proposes that our terms have to decrease progressively in size as it happened in our series located in 5.1 and 5.4. We question the validity of this assumption again. 9.2 Now we consider a series showing a somehow erratic behavior of the accumulations. We want to emphasize that checking a “few” accumulations may not suffice to predict the existence of a barrier and hence to make equality (d) in 6 valid.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
Our point is that what really matters is not the behavior of the first accumulations (as large as it number may be) but how the tails of the series behave. 9.3 Once clarified that a progressive reduction in size (not necessarily monotonic) without lower bound is a reasonable demand (since our examples are build
up from monotonic sequences, we do not want to enter in distinctions between “approaching zero” and “progressive reduction of size without lower bound”), we ask if such a demand is sufficient to ensure the validity of equality (d) and we put forward the series 1+1/2+1/3+… We start the search for a suitable barrier with disheartening results. No candidate for barrier is available and hence no way to decide on the validity of the equality 1+1/2+1/3+… = x for some x. This example shows again the limitations of our visual and computational (up to a fixed number of digits) approach to decide and hence suspicion arises that some arguments of a different nature might be necessary to ensure conclusions. We provide such a one. 10. How far can we proceed armed only with the predicting device in presence with a variety of series which may appear? On the plus side, we point out some situations where reasonable conjectures may be stated using the device. On the other hand, solid reasons are provided to justify the need of a mathematical theory in the conventional sense to be able to draw conclusions with certitude and exactness. One word of caution: visualization is a powerful tool and machines are very useful, but since they can easily deceive in extreme
situations, skepticism will be a constant companion as a way to build confidence in pure thinking. Particularly one has to concern himself with the effects of scale and the limited viewing window inherent in computer displays which create surprising perceptual misconceptions. On the other hand, representation of ideas serves as an aid to memory and as a medium for communication. We live in a perceptual world and we have to deal with things we can perceive, but if we want to think about them, we have to develop concepts. 6 Interview and students’ responses In what follows, we have assembled a single interview with a fictitious student whose answers come from a variety of students who were able to reach the interview‟s
conclusion and were heavily edited to make the material readable and to avoid clumsiness and hesitations, although we hope that what is written below is a fair representation of their clarity of thought and adaptability. The notion of infinity comes neither from observation nor from physical experience. Our brain is a finite object and it cannot contain anything infinite, although it is able to produce notions of the infinite. Since Logic does not force the incorporation of the infinite in the Mathematics (geometricians always consider finite segments, not infinite
straight lines), certain preventions on this notion are natural in our students.
Pr.: Space is usually represented as a finite segment of line. How many points are to be found in such a segment?
St.: Many.
Pr.: Yes, but how many? A finite number or …?
St.: A finite number in the sense that I can enumerate them? Yes, but a very large number of them.
Pr.: Under your supposition, is it possible to bisect a segment using a ruler and a compass?
St.: Mm …, the center point has to belong to either half … If there are an odd number of points, there is no difficulty.
Pr.: But then you cannot proceed bisecting again.
St.: Yes.
Pr.: Then is better to admit that there are not a finite number of them.
St.: But how do I know if there are a finite number of points or not in reality?
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
8
Pr.: We are not dealing with reality, whatever that is, but we are trying to model space to be allowed to
do things like bisecting indefinitely. We are trying to mimic our understanding of space based in the
system of numbers.
St.: Why?
Pr.: Because each segment associates to a number, its length, and we are good dealing with numbers. You should never under-estimate your interest and facility with numbers. By the way, under my supposition, which length can be associated to a single point?
St.: A very small one?
Pr.: A very small but a concrete number?
St.: Yes, let us say 0.000000001.
Pr.: Take a segment of unit length. How many lengths of that size can you add to reach 1?
St.: I see what you mean. No small length will do.
Pr.: Then?
St.: Under your supposition, a single point has no length.
Pr.: Please, assume my supposition as yours to proceed further.
St.: (Doubtful) If you wish …
To summarize, the admission of the infinite adapts well to our intuitions of space suggesting that any length, as small as it may be, can be subdivided. The mathematical formulation of space takes in consideration this property: (i) any segment can be bisected by means of constructions by rule and compass (ii) any length consists of points, each one
of which does not have length and (iii) those points are related to each other in the same way as the numbers in the numerical system (between two numbers there is an infinity of them). From this point of view, a mathematical segment is infinitely divisible, whereas a material cable may not be. We proceed with questions relative to how to attach a meaning to an expression such as 1/2+1/4+1/8+…, corresponding to our first considered procedure. No student
shows difficulty in predicting the next term or any further of the expression and understands the various addition signs as an accumulation of lengths. We point out that those expressions, such as 1/2+1/4+1/8+… and 1/2+1/4+1/8+…+1/64, should have different meanings and some students refuse to admit the possibility of performing accumulations indefinitely. Students, who are not adverse to perform mentally a potentially infinite procedure, understand the tail of dots as “every time I perform bisection I add the corresponding length” and, since “the procedure can be repeated as many times as desired”; they see the aforementioned expressions as dynamic processes. The equality 1/2+1/4+1/8+…=1, that is, the assignation of a number as the result of the process (that is, the dynamic process has come to a rest) is a different matter:

Pr.: Does 1/2+1/4+1/8+… = 1 makes any sense?
St.: Not much since I know how to add a finite amount of quantities, but this procedure never ends. From the figure it is obvious that, by adding, I can get as close to 1 as I want, but I shall never reach 1.
Pr.: Why is it obvious from the figure?
St.: After a few bisections a small space is left at the right side of the figure. But I can „eat‟ it up proceeding further.
Pr.: But there is always some space untouched.
St.: No, if I can proceed as long as I want, which was your assumption, uh, my assumption.
Pr.: But, if somehow we knew how to produce and end product to the procedure …
St.: And end product?
Pr.: Yes, that the procedure has been completed in some sense and as a result we got a number showing the addition of all those quantities.
St.: I don‟t see how. But if there was a way to do it, this end product would be a number smaller than 1, because I never reach the end right side of the segment.
Pr.: Suppose there is a way to obtain an end product. Let us call it x. Why do you say that x is smaller than 1?
St.: (takes a pencil and places x very close to 1) x is very close to 1, but different from 1.
Pr.: If they are different, there is a certain distance between them and hence a space untouched by our
bisecting procedure.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
St.: Right, because it never ends. (Moves x closer to 1) It does not matter how close I put x in the drawing; at same stage is reached.
Pr.: And then?
St.: Under the suppositions that the procedure never ends and that there is a way of obtaining an end product x to this procedure, x has to be 1. But there are a lot of suppositions here!
Pr.: Yes, but if you admit them, does the equality 1/2+1/4+1/8+… = 1 makes any sense?
St.: Yes, understanding 1/2+1/4+1/8+… not as a procedure but as its end product.
We want to know if something changes when dealing with a bi-dimensional setting. Thus we turn our attention to figures 2 and 3 and, admitting our two suppositions, the student does not fail in predicting the end product of both series (2 and 1/3), although the second prediction does take some time to materialize.
Pr.: Look at the picture (Figure 2). Do you grasp what is going on?
St.: The numbers in the figure stand for the areas of rectangles and the additions are a procedure. I remind myself of a children‟s game: I secure territory; I halve what is left, add it to my conquest and proceed unopposed in that way: that is 1+1/2+1/4+… stands for a step-by-step accumulation of extensions. What do you want to know?
Pr.: How much territory has been conquered.
St.: Once I stop halving, there is something left, thus the total area is smaller than 2.
Pr.: What happens if you do not stop?
St.: I can‟t do it physically.
Pr.: But you can do it mentally.
St.: If I were not to stop, no extension will be left and hence my total conquest amounts to 2.
Pr.: Are you sure nothing is left?
St.: Yes because, if after a halving of areas a small rectangle is left, the next halving takes care of part of it.
Pr.: No matter how small it is?
St.: Yes.
(5.5) Let us drop the safety net of associated figures. First some reflections on what we are doing and how to translate them with an appropriate nomenclature:
Pr.: We have been dealing with lengths and areas but we may as well have dealt with volumes. What do our previous procedures have in common? Is it essential that the problem is stated in a one-dimensional or bi-dimensional context?
St.: No, what matters is that we are dealing with numbers and a way of accumulating them.
Pr.: Right. Forgetting about our previous figures, can you state the problem we deal with from a purely numerical point of view? You may think of 1/2+1/4+1/8+… = 1, but try to make your considerations valid for any accumulation procedure. Let us start with our first supposition.
St.: (Hesitating) What I have is what you called a procedure (bisecting, adding areas) which never ends
Pr.: In numerical terms?
St.: A way of adding quantities without end.
Pr.: But not in an arbitrary way?
St.: No, because we know at any stage what we add.
Pr.: So, in absence of figures showing some kind of procedure, there is a hidden …
St.: If the procedure is obvious from staring just at the three first terms, then there is no problem. But it
worries me that without a real problem to mimic and just providing three terms and nothing else, it may be impossible to predict the following ones. That's because there is no restriction on what might come next; in any list of numbers, chosen for no reason at all, the next number can be anything.
Pr.: You are right. A question like this (showing three numbers and asking to guess the next one) is really not a math question, but a psychology question with a bit of math involved, because you are probably looking for the list the asker is most likely to have chosen - the most likely one that has a particularly simple rule. And there is no mathematical definition for that. If you just wanted a list that starts this way, but can be defined by some mathematical rule, there is a technique that lets you find an answer without guessing. This is called "the method of finite differences," and it assumes (as is always possible) that the list you want is defined by a polynomial, and finds it. Sometimes this is what the problem is really asking for, but not necessarily in our case.

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
St.: Thus, if looking at a specific number list and not knowing if it is a random list of numbers, rather than arising from a real problem, the symbol of three numbers followed by three dots is not good enough, because I do not know what I am going to add. What shall we do?
Pr.: A formula will do? Let us say I write an = 1/2n …
St.: an meaning …?
Pr.: Each term of the procedure, what you add step-by-step: for n=1, the first, for n=2, the second on so on; sub-indices stand to denote their position in the adding procedure.
St.: A formula ensures that I know what I add at any stage …
Pr.: Yes, and that the procedure does not stop is ensured by …?
St.: Letting the sub-index n run without end.
Pr.: From now on, let us call series to the procedure a1+a2+a3+… How do you write the nth accumulation?
St.: The addition a1+a2+…+an?
Pr.: Yes. What does the equality a1+a2+a3+…= x mean?
St.: Procedure equals x means that x is the end product of the procedure, that is, of the series; the number result of accumulating all the terms of the series.
Pr.: Is it not a bit confusing? Can you phrase it as “tail of dots plus something means …”?
St.: (Satisfied) Yes, it is confusing, but I think I got it: tail of dots and nothing else means procedure, that is, series; tail of dots plus number means accumulation; tail of dots and the equality sign means end product.
Pr.: Is it indifferent how many terms I write in a series if I follow them with a tail of dots? That is, are 1/2+1/4+1/8+… and 1/2+1/4+1/8+1/16+1/32+… the same series?
St.: Yes, because it is clear from the first expression which terms are going to appear in fourth and fifth position. To explicit three terms is enough, if from their expressions it is possible to deduce the next one or any subsequent one.
Once we have stated in numerical terms what a series and an end product for is, we point out that, in presence of a formula, we know what we are going to add at each step, but in absence of a figure, we do not have a geometrical understanding of what we are doing and hence no candidate for end product is available. Thus we need a way of
predicting for an arbitrary series a number which can act as its end product.
Pr.: OK, we have a series, that is, an instruction to add terms progressively and a formula saying which one is every term. Agreed?
St.: Yes.
Pr.: Moreover we know what an end product for the series is. Regarding end products, is the formula a good substitute for the figures we enjoyed before in order to make a prediction?
St.: (Checking on known series and looking puzzled) There is no apparent relation. The formula does not seem to give a clue about the end product. If, instead of picturing the procedure, you simply wrote 1/4+1/16+1/64+..., I would never have guessed that its end product is 1/3.
Pr.: Right. Perhaps the way is to start with a series, construct a geometrical situation based on it and proceed guessing …
St.: This might be very difficult, since geometry is not my forte.
Pr.: If this is not the way, then what?
St.: I do not know, but I‟ve got the feeling that you are going to tell me. Before introducing a prediction device, we need to refine our ideas and the ways of expressing them. Since the only known examples of series adding to something concrete to him come from the visualization of a procedure and since it seems not very effective to
maintain our dialog in strictly arithmetical terms, we play by his rules trying to obtain answers with mathematical content using colloquial language where terms such as “exhaust”, “distance”, “close”, “movement”, etc … appear profusely, not a bad thing since we are dealing with a dynamic process. The cleanup will come later when algebraic
maturity is available.
Pr.: In our previous examples we had a good guess for the end product. Based on what?
St.: The depletion of space in the figures provided the answer.
Pr.: How do you translate this depletion in a purely numerical way?

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
St.: The accumulations result of the procedure move forward getting closer and closer to the number suggested as end product by the figure, total length or area.
Pr.: In numerical terms?
St.: It does not matter if the terms are lengths or areas, since they are all numbers. The accumulations of those numbers get closer …
Pr.: The word “close” is vague. The planet Mars comes close to Earth when it is 90 million kilometers
away.
St.: (Smiling) No kidding! When I say that the approximations get closer to the candidate, I mean that
they are exhausting the distances between them and the end product.
Pr.: Suppose you are standing on the end product, you turn your head left and look at the accumulations.
What do you see?
St.: The accumulations keep coming near me.
Pr.: The same you would see if standing on a number larger or smaller than the end product?
St.: Well, if the number I am standing on is smaller than the end product there will be an accumulation
which passes me by.
Pr.: Only one?
St.: That one and all forthcoming accumulations.
Pr.: Let us say you are standing at the end product and you decide to step backwards, then you are
surpassed by almost all accumulations, that is, by all except a finite number of them. That will do for any
backward movement you do as small as it may be?
St.: Yes.
Pr.: What happens if you are standing at a number larger than the end product and decide to make a
backward movement?
St.: Depending on the extent of the movement. If I land behind the end product, the same as before
happens. If I move in such a way that I am still standing at the right of the end product, nothing happens.
Pr.: Meaning?
St.: I am not surpassed by any accumulation, my position acts as a kind of barrier no accumulation is
able to surpass it.
Pr: But surely not the smallest possible barrier?
St.: (Doubting) Any position at the right side of the end product is a barrier.
Now we are close to a formulation with a strong mathematical content:
Pr.: Could you say that the end product, if there is one, is the smallest barrier with this property?
St.: Yes. You said “if there is one”. It seems to me that there is a connection between end product and
barrier. No barrier, then no end product?
Pr.: It seems reasonable. Again, if a number x selected by you is the end product of a series, what should
the relation between x and the series be?
St.: x is the end product if a backward movement from x defines a zone containing all approximations
except a finite number of them, which are outside.
Pr.: Just one backward movement from x?
St.: No, all backward movements from x have the same effect …
Pr.: The finite number of accumulations lying outside …?
St.: Depending on the size of the backward movement; the smaller the movement the larger the number
of accumulations outside.
Pr.: But always an infinite number inside?
St.: Yes, always.
We summarize what has been achieved so far to ensure that we (interviewer and
interviewee) agree on the following: What we have here is a series, that is a progressive
(never ending) accumulation of quantities (our first supposition) and accepting that we can
really add all those quantities (our second supposition) we get a number which is what we
call the end product of the series. The problem is how to perform this addition: when a
series arises from a geometrical situation we are able to guess which number this end
product is going to be (based on a progressive depletion of space), but when no figure is
available we need a way of predicting the end product of a series.
If we were able to develop a prediction, than we should be able to ensure if our
guess is correct by checking if our candidate is a barrier for all accumulations and if it is
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
12
the smallest barrier, meaning that any backward movement from our candidate produces an
accumulation larger than the retreat and this has to happen for any backward movement.
Now it is time to introduce our predicting device (see 5.7) and allow the student to
spend some time familiarizing himself with both screens and operative buttons since,
according to Piaget (1977), going visual is usually more difficult and harder to teach since
one only sees what it really understands. Once the operations provided by the device have
been understood, we engage in a dialog about how to translate our one-dimensional
considerations on barriers and backward movements (to check if a certain barrier is the end
product) to a bi-dimensional setting as our screen provides explaining him that we changed
to two dimensions (screen) to perform a provide clearer visuals. Without trouble the
student identifies barrier with horizontal line and backward movement with band. Now we
have to clarify how to translate visually the “closest” number idea: draw an horizontal line
at the height corresponding to the conjecture, choose a width for drawing a band
downstairs to replicate the concept of backward movement and a message in the screen
will tell the student how many steps lie outside the band. This number will increase as the
band gets smaller in width, but always a finite amount will be outside the band and
infinitely many inside. Should this happen for every band, then the conjecture is precisely
the desired end product.
We invite him to check it with 1+1+1+… One quick look at the staircase convinces
him that in order to produce a barrier or a candidate for end product either the
accumulating terms have to be very small or have to decrease progressively in size in some
way. We discard the first possibility (5.9.1):
Pr.: How small is very small for you?
St.: Let us say 0.00…001.
Pr.: Well, consider 0.00…001+0.00…001+0.00…001+… and look at the computational screen.
St.: (Deciding on a number of ceros) Yes, we dealt with that when talking about the length of a point.
Accumulating a lot of them we can get as large a number as desired, we came across that argument
before.
Pr.: Is it possible to fill a bathtub with a spoon, time not being a problem?
St.: (Smiling) Sure!
We explore other possibilities:
Pr.: Suppose there is a barrier for a series. What would be a reasonable assumption on the terms of a
series?
St.: Each term being smaller than the preceding one?
Pr.: Is it necessary that all consecutive terms are always related this way?
St.: Well, no; we may break this rule … occasionally … I don‟t know how to express it, but globally
there has to be a decrease of value of the terms, not of the accumulations which always grow. Let us call
it a sustained decrease in size. It seems reasonable since the staircase is growing slower.
Pr.: Such as 1.1+1.01+1.001+1.0001+…?
St.: Yes.
Pr.: Please have a look at both screens. What do you see?
St.: No barrier. Mm … Oh well! This happens because each term is larger than 1 and it is essentially the
same situation as in 1+1+1+…
Pr.: Right. What is your proposal?
St.: (Hesitating) If there is a barrier, than the terms have to decrease in the sense I mentioned before,
but… not in such a way as … being all of them larger than any fixed number …
Pr..: Call it a lower bound: is this bound necessarily the number 1?
St.: Any positive number will do as lower bound.
Pr.: Let me rephrase you, existence of barrier implies terms approaching zero?
St.: (Looking puzzled) Approaching zero?
Pr: A more succinct way of putting what you said.
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
13
St.: Well … yes, existence of barrier warrantees that the terms exhibit a sustained decrease without lower
bound and that is what approaching zero means. Is it right? But if this happens, am I sure that there is a
barrier?
Pr.: No, you are not, but if a barrier is going to be found, then the terms have to decrease approaching
zero. Thus if a barrier is what you search you have to deal only with series with that property. Agreed?
St.: Yes. But what I asked is intuitively true.
Pr.: We will see.
We observe that, when asked, he looks for answers in the computational screen
more often than in the visual one, as if more trust is allocated to it. We want him to
appreciate the capabilities of the visual screen. We test our device with the series whose
end products have already been predicted 5.1 and 5.4, allowing ten digits in the
computational screen. We ask him how to conjecture if a series has an end product:
Figure 6: Visualization of 1/1!+1/2!+1/3!+…
Pr.: Our problem usually has two aspects (in these examples or others): to conjecture whether a series
has an end product or not and, if it has one, to conjecture again which one it is. Which screen do you
prefer to perform those tasks?
St.: It seems to me that the visual screen is better to predict if a series has end product.
Pr.: Why?
St.: Because with just one glimpse you can appreciate the behavior of the accumulations, that is, the
steps of the staircase … that is, if there is a tendency to go to somewhere in the vertical axis.
Pr.: This tendency will manifest itself after a few steps or after a large number of them?
St.: It depends on how fast the terms decrease. The faster the sooner I will see where the staircase is
heading to.
Pr.: As you know the factorial of an integer increases very fast. What can you predict regarding the
series 1/1!+1/2!+1/3!+…? (Figure 6).
St.: The terms decrease very fast hence I shall have a conjecture easily. Glimpsing a few steps of the
staircase I‟ll be able to guess the end product.
Pr.: What happens in a series if the terms decrease slowly?
St.: It will take more steps to see in order to guess.
Pr: For instance, consider 1/12+1/22+1/32+… (Figure 7).
St.: Yes, it takes more steps to make a guess …
Pr.: And if a series has very, very slowly decreasing terms?
St.: (Confident) A large amount of accumulations have to appear, but that is no problem, right? But, if I
ask to see many accumulations, they may collapse on the screen and I may need more than one glimpse,
but with the zoom capability the picture becomes clear again and I can decide (Figure 8).
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
14
Figure 7: Visualization of 1/12+1/22+… Figure 8: Enlarging the image.
Now we ask on how to conjecture a numerical value for an end product:
Pr.: And supposing this tendency you mentioned is present in the picture, you may conjecture the end
product?
St.: (Looking at the visual screen) Yes, approximately, in the factorial series I can conjecture 1.71 as end
product and, for the other series, 1.64 would be my guess. But I can make better guesses with the
computational screen. Guesses with better accuracy, isn‟t it? It seems that a good strategy would be to
draw as many accumulations as necessary to show the tendency and then forget about the visual screen
and activate the computational screen to conjecture a number as end product, since I only need to look to
the last accumulations to make a prediction and I can control that with the two first small windows to
show only the accumulations I want.
Pr.: How do you conjecture?
St.: When a certain block of digits repeats itself when calculating accumulations, the end product will
have that block of digits in its decimal expression.
Pr.: Hmm, thus our device provides an approximate number as end product, not an exact one.
St.: Yes, as many digits as allowed by the screen (checks with 1+1/2+1/4+… to obtain 1.999999999).
Pr.: But when we spoke about this series at the beginning of this interview you gave the exact answer 2
to our series?
St.: (Defiant) Yes, because I didn‟t use the device for it and, besides, I knew 2 to be the solution in
advance. Anyway, if I allow, let us say, twenty digits, then 1.999999999 is obviously not the end product
(he checks).
Pr.: Suppose you didn‟t know 2 to be the end product. How you decide between 2 and 1.999999999?
St.: Hmm …, the backward movement strategy?
Pr.: Yes. You haven`t checked on all backward movement away from 2. Suppose you retreat 0.0001
away from 2. Which accumulation is larger than the backward movement?
St.: (Draws a very narrow band, looks at the staircase (Figure 9) and checks in the computational screen)
For n=15, we obtain the accumulation 1.999969482, hence the 15th accumulation will do.
Pr.: What you have done for this backward movement, will do for any?
St.: (Checks further) Yes, so it seems; if the screen has digits enough to show.
Figure 9: Visualizing 1+1/2+1/4+…
In order to increase his confidence on the predictive powers of the device, we offer him
some more geometric series such as
3
5
n
n
 
 
 
 (Figure 10),
1
3
n
n
 
 
 
 (Figure 11),
8
9
n
n
 
 
 

THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
15
(Figure 12) to be studied and we ask him to conjecture end products for them and to check
his conjectures using the backward movement strategy.
Figure 10:
3
5
n
n
 
 
 
 Figure 11:
1
3
n
n
 
 
 
 Figure 12:
8
9
n
n
 
 
 

To deal with what was said in 5.9.2, we choose the series
1!/1001+2!/1002+3!/1003+… (Figure 13):
Pr.: Draw a staircase with, let us say, one hundred steps and have a look at the screens. What do you see?
St.: Convincingly heading somewhere. I prefer the computational screen: I see 1.0102 for the 100th
accumulation.
Pr.: Proceed to the 200th accumulation.
St.: Still 1.0102, which seems to be the end product.
Pr.: Just to be sure, add 60 terms more.
St.: (Looking bored) All right, 1.0108. It is almost the same as before.
Pr.: Now go for 300 terms.
St.: (Suddenly awaken) Wow! 4.59·1014. Let us check the 400 first terms: 8.54·1068. Unbelievable!
(looking at the visual screen and zooming) There is an unexpected jump up in the staircase from the
260th step on.
Pr.: Is it any lesson to be learnt from this example?
St.: There is no end product for this series.
Pr.: In general …
St.: I guess what you want is something like this: even in cases where everything looks fine in the visual
or computational screens, there is always the possibility that a change of behavior may occur lately.
Pr.: Yes. This is the first instance you have been exposed to what you may call a loss of confidence. The
device is not of straightforward application, a good nose might be helpful.
St.: Yes, but we were able to surmount the problem and we may be able to do it again in other examples.
For those situations, I find the computational screen much more informative than the visual screen. Thus
the device may work perfectly in other instances just using the computational screen.
Pr.: You seem to be very sure.
St.: Well, the problem we are dealing with is essentially arithmetical in nature and hence a large table of
values should be enough to decide.
Figure 13: Visualizing 1!/1001+2!/1002+3!/1003+…
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
16
We do not want to deal with what the term “large” may mean now that trust has
been build, an inconvenience has been eluded and faith has been restored. We offer some
reward for the trouble experienced:
Pr.: Let us see what happened in our previous example, so that you start developing a good nose. Can
you calculate some of the individual terms of the series far from the comfort zone, as you called it?
St.: (using the computational screen) Let me check 300!/100300 and some others for that matter. All big,
but if I take 10!/10010 is very small … Between 260 and 270 things begin to change …
Pr.: Yes, but you could also have deduced those considerations from the visual screen. What happens is
that the factorial grows faster than the powers of 100. Not at the beginning, where the powers are larger
than the factorial, but at some point …
St.: The factorial wins the race.
Pr.: Yes, two increasing quantities, but one of them is faster and when written as a quotient …
St.: (Interrupting) You have been cheating in this example! These terms are not getting smaller and
smaller, which was our assumption! (cheerful) That is the reason why there is no end product. What
happened here may appear in other examples, depending on the expression of an; clearly not, if an=3n/5n
as before, because both run at the same speed, so to say, and we expect regularity in its behavior.
Pr.: Any word of advice?
St.: If in doubt, one has to check further away from the comfort zone provided by the screen, where one
seems sure of the existence of an end product. The problem is how far? I may have checked hundreds of
accumulations and there is always the possibility that the series may change later. I am sure you could
have modified the expression of our last series to change behavior later.
Pr.: How?
St.: Writing in the denominator something which runs faster than 100n, I guess 1000n, but slower than
the numerator and hence taking more positions for the numerator to overrun the denominator, which is
where behavior changes in the sense that terms are becoming bigger than 1.
Pr.: Let us leave the last example at this point and state that our empirical investigation so far leads us to
believe that series with an end product have terms approaching zero. On the other hand, the only series
we know without end product are those whose terms do not behave that way. Is “approaching zero
terms” enough to ensure existence of end product?
Now to 5.9.3: it is time to create some serious discomfort regarding the device‟s
universal appeal and on our demand of ever decreasing terms without lower bound: we
present the series 1+1/2+1/3+… and we start the search for a suitable barrier (Figure 14):
Pr.: As said before, to have a reasonable conjecture of an end product for this series, we need to find a
barrier for all accumulations. Is 1 a suitable barrier?
St.: No, since the first term is 1. Since the second is 1/2, let us see if 2 is a barrier.
Pr.: Draw a horizontal line of height 2 and look for the first accumulation crossing this barrier.
St.: Clearly 2 is not a barrier.
Pr.: Set a barrier at 5.
St.: When I add 90 terms the barrier is surpassed. Well, it seems that 5 is not a barrier and the end
product is greater than 5.
Pr.: You can set a higher barrier. Set a barrier, for example, at 8.
St: (No reasonable number of accumulations chosen by the student gets him over it). Well, it seems that
8 is a barrier and hence there is end product and the end product is smaller than 8 (he even proposes a
candidate looking at the computational screen).
Pr.: A suggestion: accumulate 3000 terms and look at the computational screen.
St.: Why?
Pr.: Indulge me …
St.: (puzzled) Well, it goes over 8. Where is the trick?
Pr.: No trick. It is true that we have been accumulating very small numbers but since there are a lot of
them, each contribution is negligible and does not show in the screen (remember that we have chosen a
fixed number of digits to appear and each contribution might not be large enough to show) but its global
contribution is larger than expected.
St: Yes, again as it happened before. So it is not enough to take just a few accumulations …
Pr.: A “few” might be a very large number and still not produce any reliable conclusion.
St.: So, what to do then?
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
17
Pr.: We might need something more than our visual/computational screens, but for the time being let us
stuck to it. We still do not know what happen to our series. If the barrier is set at 15, we would have to
accumulate 2.000.000 terms in the computational screen to see that again the barrier is surpassed.
St.: Do you mean that you can always go over any barrier?
Pr.: I do not know that from the information provided by our device, but at the moment no candidate for
barrier is available and hence no way to decide on the validity of the equality 1+1/2+1/3+… = x for
some x.
Figure 14: Visualizing 1+1/2+1/3+…
Now we produce an argument to break our deadlock:
Pr.: Keep the first two terms 1+1/2 y substitute the two next ones by 1/4+1/4, from the 5th to the 8th by
1/8+1/8+1/8+1/8, from the 9th to the 16th by 1/16+….1/16 How many 1/16 are there?
St.: Eight.
Pr.: Which is the next step?
St.: (Taking its time) I see! You are dealing with powers of two. From the 17th to the 32th we substitute
the terms of the series by 1/32+…+1/32 sixteen times and so on.
Pr.: Each of the “blocks” of substitutions add to the same quantity?
St.: Yes, 1/2.
Pr.: Draw both staircases. What do you observe? (Figure 15).
St.: At every step, the second staircase is smaller than the original one.
Pr.: If there is no barrier for the second staircase …
St.: No barrier then for our series.
Pr.: Without recurring to visuals, can you argument whether there is a barrier for the second staircase?
St.: By adding 1/2 I can go as far as I want. This type of argument keeps popping up like a bad penny
Thus no end product for 1+1/2+1/3+…
Pr.: Well, we can state two conclusions: first, having a series with ever decreasing terms without lower
bound does not insure the existence of an end product for the series and secondly, our predicting device
may not be enough to proceed further.
St.: (Unhappy) It seems that way.
Figure 15: Visualizing comparisons among series.
The student manifests signs of disappointment: he made an effort to master the
device and appreciated the possibilities offered by it such as adding as many terms as
wished and the information provided by both screens showing the visual and numerical
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
18
progress of the series. Even admitting that some limitations are present (such as resolution
of the screen and zooming capabilities as well as the approximate nature of the
computational screen depending on the number of significant digits chosen) he feels
somehow cheated and demands some explanation on why such a device was presented in
the first place. One has to bear in mind that educational strategies such as the one we are
following, in spite of being interventionist, help students to construct their own
understanding of mathematics and, no matter how hard we try, it is left ultimately to the
student to decide on the adequacy of his own constructions. Once such internal agreement
is reached, it comes as a shock to have to proceed to a revision.
To placate him we study several situations where the device is useful producing
certitude or plausible conjectures, which later on will prove to be true:
(i) Comparing staircases, a comparison criterion between series can easily be stated and
understood (Figure 16).
Figure 16: Visualizing the comparison criterion.
(ii) The suppression of the first, let us say, three terms of a series with end product doesn‟t
change its character: both staircases look similar except for the fact that the second one
has been displaced a height of a1+a2+a3 with respect to the other (Figure 17).
Figure 17: Visualizing the suppression of the three first terms.
(iii) The same can be said if to a series with end product we add a finite number of terms
(Figure 18).
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
19
Figure18: Visualizing the addition of a finite number of terms.
(iv) A finite re-ordering of a series with end product maintains the same end product as
both staircases clearly show (Figure 19). It is an open problem what happens if an
infinity of terms is displaced from their positions.
Figure 19: Visualizing a finite re-ordering of a series with end product.
(v) Given a series one may introduce parenthesis that is a way of substituting several terms
by a single one, producing therefore another series. A simple look to both staircases
shows that introducing parentheses in a series with end product produces another series
with the same end product and why this happens (Figure 20 shows a series and a
second one where parenthesis have been introduced: its third position consists of the
third, fourth and fifth terms of the former series. Similar figures can be obtained by
using different choices of where to write parenthesis, not necessarily a finite number of
them).
Figure 20: Visualizing the introduction of parentheses in a series.
(vi) Plotting several staircases corresponding to generalized harmonic series leads also to a
conjecture relating exponent and existence of end product (Figure 21).
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
20
Figure 21: Visualizing generalized harmonic series.
(vii) Allowing for the possibility of having positive and negative terms in our series and
studying the behavior of the staircase in a series where addition and subtraction
alternate he easily arrives to the formulation of Leibniz‟s criterion (Figure 22).
Figure 22: Visualizing alternate series.
Pr.: Even if the device does not solve any possible situation, it has stimulated your visual perception of
what a series is and how it behaves. Let us say that the device provides a boost to your intuition.
St.: But intuition is not enough to provide precise answers to questions asked. Is that what you mean?
Pr.: The conjectural statements may be precise, but what the device does not provide is certitude, a way
of proving that those things you suspect to be true, really are true. It merely points out the direction to
follow. Conjectural reasoning as attractive as it may be is not a substitute for exact reasoning and hence
some extra tools of mathematical nature, that is algebraic-geometric-logic, need to be developed to
complement the information provided by our screens if exact answers are what we want. Somehow you
need extra arguments of mathematical nature on which to rely upon in order to cement the conquests of
your intuition.
St.: But how is it that the device sometimes fails as in our last example?
Pr.: It didn‟t fail. What happened is that you applied the device without further thought on how this
series is and you fell in the trap of predicting an end product when insufficient information had been
collected. A further application changed your conjecture on which the end product was, but after a few
tries more you ended without candidate for end product. Of course you may change your conjecture to
“there is no end product” but the device is incapable of providing reasons to decide whether there is an
end product or not and hence we needed an ad hoc argument of algebraic nature to settle the question.
St.: But your argument was very much related to the particular series we studied and probably not useful
when dealing with other series where doubt may arise.
Pr.: That‟s true and, from a mathematical point of view, is what makes the topic of series so fascinating.
Let us finish with a question: what‟s the moral of this story we have been submerged this last hour?
St.: (Vindictive) That supposing it is interesting to know if a series has end product or not, which is
something you haven‟t made clear, the moral is: either geometric simulation or the predicting device
have been found wanting in the sense that, even in those cases where predictions are possible, there is no
way of being sure if predictions hold. Very disappointing!
To avoid a possible perception of irrelevancy, we focus in his (important)
assumption: first, we point out that allowing for the infinite partition of space we are able
to model the physical idea of movement as step-by-step accumulation of lengths, that is, as
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
21
a series which, besides being a dynamic process, it has to be completed (in the sense of
calculating its end product) to conform with our modeling of reality. An incursion on
Zeno‟s paradox will clarify this idea. Secondly, we put forward the idea that when a
calculation is demanded, we or a machine are capable of performing additions and
multiplications, that is, we are good at evaluating polynomials. For other functions
emanating from geometry (trigonometric functions) or arithmetic (exponentials and
logarithms), their evaluation depends on the existence of polynomials which can be as
close as we want (depending on the accuracy required) to them, which is essentially
“adding up” a series whose terms are monomials. Last, but not least, and proceeding the
other way round, series (known to have an end product) are the most powerful tool in
creating new functions, enriching the catalogue of known functions.
Pr.: You have been studying Mathematics for quite a while now and you are used to more certainty in
the conclusions to which you arrive. How does that compare with our efforts?
St.: Mm … I haven´t made my mind yet. I don‟t want to be rude but I came here under the impression
that we were going to engage in a mathematical experience, as you advertised it. It troubled me from the
beginning that you use common language, you rely in pictures and guessing, a lot of blah, blah, and no
definitions, theorems, you know, mathematical stuff…
Pr.: No offence taken. People do not communicate ideas algebraically in everyday life; ideas only take
shape when they are put in words and sentences. What is wrong with that? With respect to reliance on
pictures, I think you can be taught to manipulate pictures as easily as you manipulate symbols. Do you
think mathematics is about language and symbolism or more about communicating ideas and providing
conjectures?
St.: I guess is more about ideas. Concerning conjectures, I do not know; it is not very often that we are
asked what our conjectures are about anything mathematical. To be honest I‟ve been thinking more
intensely along this experience than I usually do in the classroom. As a matter of fact, I feel a bit tired,
sorry …
Pr.: No need to be, you are right. Although a certain amount of symbolism was unavoidable, I just
wanted to know how far you can progress in absence of standard mathematical language and you have to
agree that we have collected quite a lot of information and, what is more important, a kind of taste or gut
feeling for the subject. But it was also my intention to create some uneasiness on you to force you to
admit that there is no shortcut eluding mathematical reasoning: once a basic grasp is achieved, we need
refinement and more systematic articulation through algebra which is supposed to be the lingua franca of
college mathematics courses; we use it precisely to avoid the vagueness of words.
St.: And all that leaves us where?
Pr.: Next week at the classroom, where I shall develop a theory on numerical series based on a
definition: I shall call convergent series to a series with end product and its definition will be a
translation in logical-algebraic terms of the idea of proximity between accumulations and end product
you established in our dialog. Then I„ll rephrase this definition just involving the terms of the series and
not the end product, which will liberate me of the tyranny of having to know which number it is …
St.: And hence no need for the device …
Pr.: Sure, in the logical development of the theory there is no need. You soon will learn that in this
theory there are two aspects to deal with: to know whether a series is convergent and, supposing it is,
which one is its end product. The second aspect will prove notably harder than the first, except when
dealing with certain known series. In any case, the device is useful in producing the statements I have to
prove or disprove along the construction of the theory in our first aspect and provides approximate
conjectures to the second, where the mathematical treatment for exactness may be harder and outside
your present reach.
7 Final Words
In Calculus courses, the student is asked questions about dynamics of the „real-world‟
situations that requires extracting dynamic information from a static graph. Surely it ought
to be easier to extract dynamic information from a dynamic presentation. The generation of
images promotes the integration of the separate components of the item in question and
accessing parts of the information encoded in memory prompts the retrieval of all other
pieces of information contained in the image.
THE INDEFINITE ACCUMULATION OF FINITE AMOUNTS
22
To investigate convergence of series, a computer-generated tool based on a powerful
mathematical assistant was designed as a Graphic User Interface (GUI) allowing the
combination of multiple windows (graphic and computational) with interactive capabilities
to show different representations of the concept and simulate dynamism. It is clear to us
that its use may complicate learning from a didactic point of view, since it is mainly a
materialization of symbolic technology and hence changes the material to be taught by
transposing everything to a computational problem, but its careful use can provide a richly
textured view of the problem in question. On one hand, left alone, adverse effects in
creativity and problem solving are bound to arise, putting the student in extreme
dependence to the machine, anesthetizing his impulses to look for other representations of
the problem. On the other hand, as said, we see only what we understand and to address
potentially damaging effects, the use of the GUI needs the company and the complicity of
a well-structured dialog to ensure success.
The interview was not a wild-goose chase. As a general rule, an interview has to be
carefully planned with a strategy in mind and orderly implemented, avoiding inappropriate
tactics at the wrong times or places and allowing for feedback to planning and assessment.
Plenty of deliberate silences have to be planted along the interview to cause a flurry of
thought activity. In our concrete case, its design had to comply with the educative model
chosen and should be instrumental in detecting the specified levels of progress in
understanding. In order to respect point (3) as specified in Section 3, the interview induced
students to talk profusely allowing us in to analyze those slight changes in their use of
language which the model postulates as indicative of progress; to comply with point (4),
computational skills were left to the GUI in order not to contaminate our appreciations.
Those students which reached the end of our interview sailed through cycles of resistance,
acceptance, certainty, and loss of it and, at the end, admission of the need of more
sophisticated tools of mathematical nature. It seems to us that this approach is the best way
to proceed prior to the establishment of a mathematical theory in the classroom and not
only for weaker students. Of course not all interviewees were capable of completing the
interview; as a matter of fact, only one third of them finished it satisfactorily. It was not
our purpose to leave the job of analysis of data to the reader, but extension considerations
lead us to present only our empirical work and leave for a forthcoming article how it all fits
within van Hiele‟s model, why certain students failed to reach the higher levels of the
model, how we extended our study to a larger sample of students via a multiple choice test
designed as a spin-off of the interview and why such a limited degree of success was not
unreasonable. For those readers more interested in the methodological aspects of our
strategy than in its theoretical foundation, this article may stand on its own by providing
hints on how to proceed individually with students or even on how to adapt its contents as
classroom material. By request we can provide the interested reader with the design of our
predicting tool.
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