Conjecturing in a computer microworld: zooming out and zooming in.Teachers and researchers agree that making and testing conjectures This is an incomplete list of mathematical conjectures. They are divided into four sections, according to their status in 2007. See also:
NCTM Nationally Certified Teacher of Music NCTM North Carolina Transportation Museum NCTM National Capital Trolley Museum NCTM Nationally Certified in Therapeutic Massage , 2000). Conjecturing--activity of making, testing and interpreting conjectures--is seen as a major pathway pathway /path·way/ (path´wa) 1. a course usually followed. 2. the nerve structures through which an impulse passes between groups of nerve cells or between the central nervous system and an organ or muscle. to discovery, and is thus indispensable in problem posing and solving. As such, teachers are encouraged to support students' conjectures through collaborative verification, rather than treat them as statements that prompt haste judgment. In this article we focus on conjectures and conjecturing in computer-based environments. Some researchers have argued that such environments can be particularly supportive of student conjecturing by providing empirical access to mathematical properties and relationships (Cuoco & Goldenberg Goldenberg may refer to: People:
adj. Calling for, requiring, or showing little or no effort. See Synonyms at easy. ef  fort·less·ly adv. gather greater amounts of
reliable data, in which patterns may be more easily discerned. In
addition, many computer environments are more powerful than calculators,
and capable of working flexibly with larger sets of data, thus offering
alternative and multiple representations of information. As such,
various kinds of patterns and relationships are made more accessible,
depending on the representations chosen. Most importantly Adv. 1. most importantly - above and beyond all other consideration; "above all, you must be independent"above all, most especially perhaps, well-designed computer-based environments can be created to provide learners with instant, pedagogically ped·a·gog·ic also ped·a·gog·i·cal adj. 1. Of, relating to, or characteristic of pedagogy. 2. Characterized by pedantic formality: a haughty, pedagogic manner. useful feedback, frequently visual in nature, that can be used by learners to evaluate or reflect on their own actions (Goldenberg, 1989; Noss & Hoyles, 1996). As a result, some learners adopt a greater sense of agency in their mathematical learning as authority is transferred from the teacher to the computer (Hewitt Hewitt may refer to:
Our goal in this article is twofold. First, we take a broader view, "zoom To change from a distant view to a more close-up view (zoom in) and vice versa (zoom out). An application may provide fixed or variable levels of zoom. A display adapter may also have built-in zoom capability. out", of conjecturing and situate sit·u·ate tr.v. sit·u·at·ed, sit·u·at·ing, sit·u·ates 1. To place in a certain spot or position; locate. 2. To place under particular circumstances or in a given condition. adj. it within the larger cluster of mathematical activities with which it interacts. We thus aim to provide a better account of the motivations for the effects of conjecturing in student mathematical inquiry than is currently available in the research literature, where conjecturing is frequently seen as a precursor precursor /pre·cur·sor/ (pre´kur-ser) something that precedes. In biological processes, a substance from which another, usually more active or mature, substance is formed. In clinical medicine, a sign or symptom that heralds another. to proof. Second, we examine the details, "zoom in", of the student conjecturing in the context of a specific computer-based problem situation. We examine particular elements of participants' interactions with the problem, including the triggers that shaped their conjectures and the ways in which conjecturing guided their problem solving problem solving Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. . In addition to providing a fine-grain analysis of student conjecturing, "zooming in" allows us to understand better the role a teacher can play in supporting productive conjecturing in computer-based environments. Situating This Report Participants and setting This report is part of a larger study that investigates the interaction of a group of preservice teachers with a web-based microworld referred to as "Number Worlds" (the microworld is introduced below). The study investigated the mathematical experiences and understandings of 90 preservice elementary school elementary school: see school. teachers who interacted with the microworld using a set of given mathematical tasks. These tasks provided the participants with the opportunity to explore concepts related to elementary number theory, such as factors, divisors, divisibility di·vis·i·ble adj. Capable of being divided, especially with no remainder: 15 is divisible by 3 and 5. di·vis , prime numbers There are infinitely many prime numbers. The first 500 are listed below, followed by lists of the first prime numbers of various types in alphabetical order. The first 500 prime numbers 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 , and reexamine re·ex·am·ine also re-ex·am·ine tr.v. re·ex·am·ined, re·ex·am·in·ing, re·ex·am·ines 1. To examine again or anew; review. 2. Law To question (a witness) again after cross-examination. their understanding of these concepts in an environment that supported experimentation and visualization Using the computer to convert data into picture form. The most basic visualization is that of turning transaction data and summary information into charts and graphs. Visualization is used in computer-aided design (CAD) to render screen images into 3D models that can be viewed from all . A comprehensive description of various aspects of this study is found in Sinclair, Zazkis and Liljedahl (2003). In this article we focus only on 20 participants (out of 90), who volunteered to participate in a clinical overview about their Number Worlds experiences, and on one mathematical problem Mathematical problem may mean two slightly different things, both closely related to mathematical games:
Description of the Number Worlds microworld Figure 1 shows a snapshot (1) A saved copy of memory including the contents of all memory bytes, hardware registers and status indicators. It is periodically taken in order to restore the system in the event of failure. (2) A saved copy of a file before it is updated. of the Number Worlds applet A small application, such as a utility program or limited-function spreadsheet or word processor. Java programs that are run from the browser are always known as applets. See midlet, crapplet and Java applet. . The centre grid contains a two-dimensional array of clickable clickable adj (COMPUT) → cliqueable clickable adj → cliccabile cells. The numbers shown in the cells depend on the world that has been chosen. Although the basic objects of Number Worlds are the positive integers, the user can define a custom subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. of objects: the Natural World, Whole World, Even World, Odd World, and Prime World. Each world displays the corresponding set of numbers. It is also possible to change the numbers shown in the cells. The user can increase or decrease the start number by one row, that is, by the value of the grid width, or simply by one cell. The Reset Grid button resets the start number of the current world. [FIGURE 1 OMITTED] The appearance of the grid can be affected by changing the value of the Grid width menu. By selecting values from one to twelve, the user can change the number of columns displayed, and thus the total number of cells. There are always exactly ten rows. Within each world, the user can highlight, Show, certain types of numbers: Squares, Evens, Odds, Primes, Factors, and Multiples. Further, the Show Multiples can be accompanied with Shift them by an integer integer: see number; number theory in order to create various arithmetic sequences. Finally, the four basic arithmetic operations are available to the user. The result of the operation, in addition to the two inputs, is highlighted on the grid while a string representing the operation and the result appears above the grid. Some mathematical ideas of Number Worlds relevant to this study Elementary number theory is concerned with the structures and relationships of natural numbers. Therefore, in designing the Number Worlds microworld, we chose to focus primarily on this set of numbers. However, instead of the one-dimensional one-di·men·sion·al adj. 1. Having or existing in one dimension only. 2. Lacking depth; superficial. one-dimensional Adjective 1. having one dimension 2. number line representation traditionally used, we adopted a two-dimensional grid display, thus maximizing the real estate of the screen. The two-dimensional grid display also sheds a different light on the relationships among the numbers and provides an opportunity to construct or reconstruct re·con·struct tr.v. re·con·struct·ed, re·con·struct·ing, re·con·structs 1. To construct again; rebuild. 2. these relationships. In addition, it offers a different external representation of both factors and multiples, as well as primes and square numbers, in ways that provide concrete visual instantiations of algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind. [CACM 2(5):16 (May 1959)]. 2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. relationships. Further, by producing unexpected patterns, the two-dimensional grid display offers a novel representation of the numbers, which we hoped would provoke pro·voke tr.v. pro·voked, pro·vok·ing, pro·vokes 1. To incite to anger or resentment. 2. To stir to action or feeling. 3. To give rise to; evoke: provoke laughter. surprise and engagement for our research participants. In what follows we provide several examples of mathematical relationships inherent in Number Worlds. Example 1: Multiples and Stripes The relationship between the input show multiples of and the grid width determines the type of pattern. For example, in the Natural World, showing multiples often on a grid width often results in one highlighted column (see Figure 2) referred to as a stripe stripe - data striping . Showing multiples of five or two on the grid width of ten results in two or five highlighted stripes, respectively. In general, a pattern of stripes emerges when the input to show multiples of is a factor of the grid width. [FIGURE 2 OMITTED] Example 2: Multiples and Diagonals. As shown above, a stripe pattern emerges when the input to multiples m is a factor of the grid width n. However, this is not the only way to produce an aesthetically appealing pattern. Figure 3 shows multiples of 2, 4, 5 and 7 on a grid width of 9. We refer to the patterns in (b) and (c) as diagonals, and refer to the pattern in (d) as disconnected diagonals. The pattern in (a) can be seen as diagonals as well, however, it is more naturally identified as a checkerboard checkerboard the pattern of a chess or draft board; used in many circumstances to display the results of mixing a specific number of variables. The variables are listed in columns designated along the horizontal border and the same or different variables in lines along the vertical pattern. [FIGURE 3 OMITTED] These images are examples of the following general relationship: diagonals are displayed by multiples of m on a grid width of n, where m is a factor of n [+ or -]1, and disconnected diagonals are displayed by multiples of m on a grid width of n, where neither n nor n [+ or -]1 are a multiple of m. Checkerboards are a special case of diagonals and they are displayed by multiples of two on any odd grid width. (This precludes the creation of a "real" checkerboard pattern of width eight.) These patterns hold for any shift value; increasing the value of the shift will simply move the pattern over. By stacking the common differences on top of each other, the "commonness" of the differences becomes visually obvious even when diagonals, rather than columns, are being displayed. We attend to this feature below in our description of the research participants' work with the microworld. Example 3: Patterns shifted Previous examples discussed the visual pattern that emerges as a result of the relationship between the input to show multiples of and the grid width. Another value that plays a role in the visual display is the input to shift by. It does not alter the pattern, but translates it accordingly. Figure 4 displays the result of shifting multiples of 5 by 3 and (-3) on different grid widths. [FIGURE 4 OMITTED] In general, by showing multiples with a shift, the user can generate any arithmetic sequence. The input for show multiples of corresponds to the common difference while the initial term is the sum of the inputs to show multiples and shift by in the Natural world. The problem and mathematics behind it In this report we focus on the following mathematical problem that was presented to participants in a clinical interview setting: The grid width is 9, with multiples of 4, shift 2 highlighted. The first number in the grid In the Grid is a game show that airs on UK broadcaster Five at 6.30pm week nights. It first aired on Monday 30 October 2006. In the Grid is hosted by Les Dennis and is produced by Initial West, one of the Endemol UK companies. is 190. How else could we highlight the same numbers? Figure 5 shows the grid display according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. the requirements of the problem. The width of the grid has no influence on the highlighted numbers; it affects only the pattern of display. The same display will result from showing multiples of 4 shifted by 6, 10, 14 etc., or, in general, by a shift of 4k+2 for integers k. (In fact, these integers k must satisfy k < 47. Otherwise the input for shift will exceed 190 and jeopardize jeop·ard·ize tr.v. jeop·ard·ized, jeop·ard·iz·ing, jeop·ard·izes To expose to loss or injury; imperil. See Synonyms at endanger. the requirement of the "same numbers shown." We note this respecting mathematical accuracy, rather than claiming the relevance of this observation to the participants' work.) Further, change of the input to the shift is the only way to generate the same numbers, as any other input to show multiples of will result in a different spread of the numbers. [FIGURE 5 OMITTED] Prior research has shown that preservice teachers' understanding of multiples is often disconnected from the property related to the distribution of multiples of n among the natural numbers, namely, that every nth number is a multiple of n (Zazkis & Liljedahl, 2002). In designing this problem, we were therefore interested in whether the participants had, or could achieve and implement this understanding. Further, Zazkis (2001) has noted that inviting learners to consider "large" numbers helps divert di·vert v. di·vert·ed, di·vert·ing, di·verts v.tr. 1. To turn aside from a course or direction: Traffic was diverted around the scene of the accident. 2. their attention from specific numbers to pattern and structure. Hence, we chose to set the first number on the grid at 190. Though 190 may not be seen as large, the sequence of numbers that starts with 190 does not draw upon the immediate computational Having to do with calculations. Something that is "highly computational" requires a large number of calculations. capabilities of the participants. For this report, we have chosen to analyze this question in detail for interrelated in·ter·re·late tr. & intr.v. in·ter·re·lat·ed, in·ter·re·lat·ing, in·ter·re·lates To place in or come into mutual relationship. in reasons: it appears to be the most problematic for our participants and, as a result, they engaged in a significant amount of conjecturing and experimenting. Their different approaches are of specific interest here as they illuminate il·lu·mi·nate v. il·lu·mi·nat·ed, il·lu·mi·nat·ing, il·lu·mi·nates v.tr. 1. To provide or brighten with light. 2. To decorate or hang with lights. 3. the process of conjecturing in a problem solving situation and provide a context in which this process is examined. Zooming Out: Students' Experimental Approaches The NCTM Standards describe a conjecture CONJECTURE. Conjectures are ideas or notions founded on probabilities without any demonstration of their truth. Mascardus has defined conjecture: "rationable vestigium latentis veritatis, unde nascitur opinio sapientis;" or a slight degree of credence arising from evidence too weak or too as an "educated guess." In contrast, we take a broader view on conjectures in computerized computerized adapted for analysis, storage and retrieval on a computer. computerized axial tomography see computed tomography. settings, and situate conjecturing as one of several experimental approaches used in mathematical problem solving. We use the term experimental approaches to describe ways of discovering results that rely on empirical methods Empirical method is generally taken to mean the collection of data on which to base a theory or derive a conclusion in science. It is part of the scientific method, but is often mistakenly assumed to be synonymous with the experimental method. . Experimental mathematicians Mathematicians by letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also
tr.v. ver·i·fied, ver·i·fy·ing, ver·i·fies 1. To prove the truth of by presentation of evidence or testimony; substantiate. 2. analytically-derived results; (2) discovering new patterns and relationships; (3) testing and falsifying fal·si·fy v. fal·si·fied, fal·si·fy·ing, fal·si·fies v.tr. 1. To state untruthfully; misrepresent. 2. a. conjectures; (4) gaining insight and intuition intuition, in philosophy, way of knowing directly; immediate apprehension. The Greeks understood intuition to be the grasp of universal principles by the intelligence (nous), as distinguished from the fleeting impressions of the senses. (Borwein & Bailey, 2004). During the clinical interviews, we gave the participants access to Number Worlds in order to determine whether they would (and could) use it to help them solve problems. The range of use went from minimal to complete reliance. Though our participants depended on experimental methods, they seemed to do so for different purposes. We wish to examine whether the purposes of experimental methods used by mathematicians, described above, can guide our analysis of students' experimentation. We thus attempt, in the three following cases, to identify the experimental methods that appear in students' interaction with the problem. Case 1: Celia
Interviewer: Is there any other way that you could get the exact same
numbers highlighted?
Celia: So multiples of 4, shift them by 2, 90, 94, 98, 202, 206,
okay going up by 4's, you're showing also even numbers, but
if you show even numbers, you'll get every second one, and
don't want that ... Oh okay, (pause) multiples of 4, um,
what about (pause) okay, I'm thinking multiples of 4 will
have to give you everything in 4's, so that's what you
need, so you can't really change that, but you can change
something else to shift it back, or shift it forward ...
[...]
Interviewer: Feel free to experiment, you don't have to get the answer
right away ...
Celia: Yeah, (pause) multiples of 4, shifted by 2, I'm trying to
think, because we have to go in 4, there isn't any other
way to express that. Show primes, show squares, evens,
factors, so factors of (pause) on here so, show multiples
of 5, it'll go every 5th one, no it won't work. Multiples
of 2, every second one, multiples of 4, I don't really
know, um, (pause) can I use a piece of paper to ...
Interviewer: I'm just going to comment on what you do, you're writing
down the multiples of, are you writing the 190, 194,
198 ...
Celia: Yeah, 202, so the real multiples of 4 shifted by 2, are
188 ... 192, 196, 200, so these are the multiples of 4,
(pause) ... So whatever relationship is multiples of 4,
like multiples of 2, multiples of 8 have a relationship to
multiples of 4, but if you say multiples of 8, you won't
get every cell, if you say multiples of 2, you get twice as
many cells, so ...
Interviewer: So how do you get from there to there, from that column to
this column?
Celia: Well you shift 2, you add 2, so I'm trying to think of
another relationship of numbers to each other that would
highlight every 4th one and then change what you would do
to get there, to get to the original highlighted cell, but
nothing is coming to mind that the same relationship is
multiples of 4 ...
Interviewer: Well I think you're right, I mean you said it earlier, we
have to stick with multiples of 4.
[...]
Interviewer: So how could you get from this to this?
Celia: You would shift 2, which we're already doing, multiples of
4 shifting 2 ... (pause) Oh minus 2, (pause) shift negative
2 ...
Interviewer: You want to try that?
Celia: Sure. So we want to start at 186, no we want to start on
umm okay, well let's just start, let's just try it from
shifting negative 2, see what happens ...
In this excerpt ex·cerpt n. A passage or segment taken from a longer work, such as a literary or musical composition, a document, or a film. tr.v. ex·cerpt·ed, ex·cerpt·ing, ex·cerpts 1. we see Celia's attempt to derive her answer analytically an·a·lyt·ic or an·a·lyt·i·cal adj. 1. Of or relating to analysis or analytics. 2. Dividing into elemental parts or basic principles. 3. . She first concludes that the input to "multiples of" should stay unchanged, however, she then doubts this conclusion and tries to find another relationship that will result in highlighting every 4th number. Despite the interviewer's encouragement to "feel free to experiment, you don't don't 1. Contraction of do not. 2. Nonstandard Contraction of does not. n. A statement of what should not be done: a list of the dos and don'ts. have to get the answer right away," Celia insists on thinking through the problem analytically. Her question, "Can I use a piece of paper?" announces this tendency. She needs the interviewer's guiding prompt, "So how could you get from this to this", in order to suggest a shift of -2. Following further suggestion of experimentation by the interviewer, "you want to try that", Celia agrees. Her words, "Well let's let's Contraction of let us. just start, let's just try it from shifting negative 2, see what happens" here indicates that she is inputting -2 in order to gain some insight and intuition from the result, rather than to test a conjecture or an analytically derived result. "Just start" could indicate her mental switch from thinking/analyzing to experimenting. Case 2: Jake JAKE Jointly Administered Knowledge Environment
Interviewer: Okay we're going to show the multiples of um 4, shift 2.
And this time the question is, um, how could you use a
different combination there to highlight exactly those same
numbers?
Jake: (pause) I was going to try multiples of 2 and a shift of 4
just because I want to see what it does, I have no reason
to believe this would work.
Interviewer: Okay ...
Jake: Obviously not, what am I thinking.
Interviewer: That's okay, you've got to start somewhere. So you've got
it back now at multiples of 4, shift 2. Why did you say
obviously, when you saw ...
Jake: Because it was 2 on a grid width of 9 so it showed a
checkerboard, which was obvious ...
Jake: I'm making shift for starting at 0 (laugh), um, I'm trying
to think of what the relationship is.
Interviewer: So what is it? Oh so you're doing shift of 8, (pause) shift
of 7, shift of 6
Jake: There we go ...
Interviewer: Good, but why was 8 supposed to work?
Jake: Well I was thinking, I don't know why, it was a guess
because obviously there is a difference of 2 between the
shift and the multiples of 4, basically you're just moving
the shift over, but it's the same thing ...
Interviewer: Right okay, so what else would work besides 6?
Jake: 10, 14 ...
Jake's initial choice of input is based on his intention to "see what it does;" he doesn't does·n't Contraction of does not. expect this will give him the desired result. However, upon consideration of the resulting display, which he refers to as "obvious", he returns to multiples of 4 and systematically tries different inputs to the shift. This systematic approach can be seen as testing series of conjectures or as a deliberate search for a pattern, which, indeed, is found and explained as a result of a successful input. His repeated use of "obviously" in a reaction to what the computer displays suggests that he came to understand the pattern as a result of his experiments. Case 3: Alice Alice, city (1990 pop. 19,788), seat of Jim Wells co., S Tex.; inc. 1910. Long a cow town at a railroad junction, Alice remains a cattle-shipping center. Oil and natural gas are also important to its economy. Manufactures include office equipment and fishing tools.
Interviewer: My question is, is it possible by giving different input to
highlight exactly the same numbers?
Alice: Keep that 190, um, maybe a shift of 6 ...
Interviewer: Maybe a shift of 6, and why 6?
Alice: 4 + 2 is 6 ...
Interviewer: Okay, let's try ... So shift of 2, shift of 6, gives us the
same pattern. Okay, what else can we put there instead of
6?
Alice: Okay, let's try another even number, so 8 ...
Interviewer: 8?
Alice: 8, [sees result] maybe not, let's try now another even, 12
Interviewer: Let's try, okay ...
Alice: Let's see what 13 does ...
Interviewer: And how are you choosing those, I understand you have
chosen 8, then 12, and you said let's try another even
number, so what is with 13?
Alice: 4 + 9, I wanted to see what that would do ... It gets most
of the same numbers, oh no ...
In the beginning it appears that Alice has a definite conjecture in mind. Her input of 6, based on coincidental co·in·ci·den·tal adj. 1. Occurring as or resulting from coincidence. 2. Happening or existing at the same time. co·in sum of inputs 4 and 2, brings a desirable result. From here Alice's experiments seem to be based on a conjecture that all even input for the shift will result in the same display. However, this conjecture is falsified for 8 and then for 12. At this point Alice abandons her conjecture and tries a shift of 13, which is the sum of two of the inputs, 9 and 4. Alice's explanation that accompanies this experiment, "I wanted to see what that would do", hints that, at that moment, she may not be testing conjectures any longer, but trying to gain insight and intuition. Experimentation and conjecturing Though we do find in students' work experimental approaches similar to those that have been identified in the work of mathematicians, the distinction at times is difficult to make. Is Celia searching for an analytical analytical, analytic pertaining to or emanating from analysis. analytical control control of confounding by analysis of the results of a trial or test. result to be verified ver·i·fy tr.v. ver·i·fied, ver·i·fy·ing, ver·i·fies 1. To prove the truth of by presentation of evidence or testimony; substantiate. 2. or is she trying to come up with a conjecture? Is Jake searching for a pattern or is he trying series of conjectures? Does Alice really just try to see what is going on, or does she have a conjecture that she doesn't share with the interviewer? The boundaries are blurry blur v. blurred, blur·ring, blurs v.tr. 1. To make indistinct and hazy in outline or appearance; obscure. 2. To smear or stain; smudge. 3. : When a student has a strong faith in correctness of a conjecture, or a reasonable justification for this educated guess, conjecturing may appear as verification of a result that was derived analytically. When a student puts her conjecture in doubt, conjecturing resembles gaining intuition and searching for patterns, or, in students' words, "just trying." Even though the actual motivations behind the participants' experimentation are at times vague, we note that each experimental attempt, even the naive naive - Untutored in the perversities of some particular program or system; one who still tries to do things in an intuitive way, rather than the right way (in really good designs these coincide, but most designs aren't "really good" in the appropriate sense). looking ones, carries some intent, whether it be to test a conjecture, gain intuition, search for patterns, verify (1) To prove the correctness of data. (2) In data entry operations, to compare the keystrokes of a second operator with the data entered by the first operator to ensure that the data were typed in accurately. See validate. a result, or some combination of these. While mathematicians may be able to monitor their intents and motivations in their own mathematical experimentation, it may be that students need to develop this awareness in order to capitalize on Cap´i`tal`ize on` v. t. 1. To turn (an opportunity) to one's advantage; to take advantage of (a situation); to profit from; as, to capitalize on an opponent's mistakes s>. the experimental possibilities offered by the computer. Zooming in on Students' Conjectures Although the participants displayed elements of each of the four experimental approaches outlined in the previous section, the majority of their experiments can be viewed as testing conjectures, regardless of whether these conjectures are stated explicitly or are implicit in Adj. 1. implicit in - in the nature of something though not readily apparent; "shortcomings inherent in our approach"; "an underlying meaning" underlying, inherent students' choices. As such, in this section we examine the participants' conjecturing in further detail. In particular, we examine a variety of inputs chosen by students, the issues that guided their choices of input, and their interpretations (or lack of interpretations) of the output offered by Number Worlds. We note that all the interviewees eventually solved the problem. That is, they all reached the conclusion that the input to show multiples of should remain unchanged while the input to shift be drawn from the set {... -6, -2, 2, 4, 6, 10,...}. Participants varied in their understanding of this set of inputs. Some could only suggest specific examples of inputs in the set that fit the pattern. Others were able to describe elements of this set as "multiples of 4, plus 2". More explicit explanations included a reference to "hitting in between" two multiples of 4. However, only six interviewees kept the input to multiples constant from the very beginning. The others found this strategy only after interpreting one or more outputs. We present here a few popular approaches that participants implemented in making and testing their conjectures. Analyzing the situation Several students started by analyzing the situation. The following excerpt provides an example of such an analysis. (It is also an example that shows that a distinction between testing conjectures and verifying analytically-derived results is-vague).
Amber: Yeah I'm thinking, multiples of 8 wouldn't work because
then it would be too many spaces, so I thought at first if
you do multiples of 8 and then shift them by negative
(pause), and 2 wouldn't work because that would be too many
numbers, hmm, (pause) just these numbers and no other
numbers?
Interviewer: Yeah, I would like to have the same numbers, 190, 194, 198,
only those numbers, but is there another possibility of
getting them, rather than multiples of 4 shifted by 2 ...
Amber: Will shifting by negative 2 also work?
Interviewer: What do you think?
Amber: Let me think about it, well, you can't change the multiple,
right, because or else that would change everything, we
have 194 is a multiple of 4, shifted by 2 and the actual
multiple of 4 is 192, so if you shifted at -2 ... I think
it would, it does.
Interviewer: This is an interesting idea, why do you think it will
change the multiple ...
Amber: Because there always has to be 3 spaces in between, or like
4 jumps to the next number, there's a multiple of 4 no
matter what we shift it by.
Interviewer: Okay, so you convinced me. Let's not change the multiple,
let's change the shift ...
Amber: Let's change the shift, maybe negative 4 and negative 6, or
even 4 or 6 ...
Interviewer: So what would you like to try, 4 or 6?
Amber: Um, try 4 ... Not the same ones, uh try 6, those are the
same ones right ...
Interviewer: Okay, we tried 4, we didn't get the same ones, we tried
shifting by 4, didn't get the same ones, now shifting by 6
did give us the same numbers ...
Amber: Okay, so 2 works, 6 would work, 10 would probably work
then ...
Interviewer: How probably?
Amber: Because they're going 4 spaces each time, like I know it
inside of my head, okay let me think about this, okay if 2
works and 6 works, we're going 4, 4 spaces, so 6 + 4 would
work and 6+4+4 would work and 6+4+4+4 ...
Interviewer: So you give me an example of another number.
Amber: Try 10, 14.
Attending to "too many spaces" in showing multiples of 8 and "too many numbers" in showing multiples of 2, Amber concludes: "You can't change the multiple." She further suggests -2 as a solution. However, in her search for other inputs she tries a shift of 4 followed by a shift of 6. This last experiment lets her see the general structure of possible solutions. However, her general structure is additive additive In foods, any of various chemical substances added to produce desirable effects. Additives include such substances as artificial or natural colourings and flavourings; stabilizers, emulsifiers, and thickeners; preservatives and humectants (moisture-retainers); and , without an attempt to generalize generalize /gen·er·al·ize/ (-iz) 1. to spread throughout the body, as when local disease becomes systemic. 2. to form a general principle; to reason inductively. the repeated addition any further. Focus on a specific number Several participants chose to focus on one specific number from the highlighted set and consider in what way this number can be built as a combination of multiples and a shift.
Interviewer: It is to get the same numbers highlighted by plugging in
different inputs here, instead of 4 and 2?
Kori: Show multiples of 3, shift by 1?
Interviewer: Do you want to try it?
Kori: Sure ...
Interviewer: And what makes you just make the suggestion?
Kori: Oh I'm just looking at the first row there ...
[...]
Interviewer: Stripes. So 3 shifted by 1, we get stripes. And we want 190
and then 194, 198 and so on.
Kori: I thought multiples of 5, shifted by negative 1, but then,
it works there [points to 194], but again further it
doesn't work.
Interviewer: 5 shifted by negative 1, let's look at how it works. Maybe
it will give you some idea of how to proceed. What made you
try 5 and negative 1?
Kori: Just looking at the first number that was highlighted, or
the second one I guess, the 194, I just figured that the
195 is a multiple of 5, and if you shift if by negative 1
it highlights the 194 ...
Interviewer: And you did, you got 194 highlighted. But we wanted 194 and
slightly different partners highlighted.
In this excerpt Kori explains that her attention is on the numbers in the first row. She first notices that 190 is one more than a multiple of 3, so she shows multiples of 3 shifted by 1. This indeed highlights the number 190, but gives a pattern of stripes. So Kori focuses on the second number in a row: 194. She recognizes it as a result of moving one unit left from 195, the closest multiple of 5, therefore her next input is to show multiples of 5 shifted by -1. Similarly to her previous attempt, she gets the number she focused on highlighted, but not the other numbers. Focus on a specific number can be seen as specializing, an approach that is discussed in the mathematics education literature as helpful especially in the initial steps of solving a problem. (Mason, Burton & Stacey, 1982). However, the ultimate goal of specializing is to engage in the process of generalizing, that is moving from the consideration of a specific case to a wide class of cases. However, Kori's attempts did not allow her to make any conjectures in the spirit of generalizing. In analogy analogy, in biology, the similarities in function, but differences in evolutionary origin, of body structures in different organisms. For example, the wing of a bird is analogous to the wing of an insect, since both are used for flight. to "over-generalizing," which serves as a diagnosis of various mathematical mistakes, we believe it is appropriate to describe Kori's approach as "over-specializing," that is, focusing on features that are so particular that they preclude pre·clude tr.v. pre·clud·ed, pre·clud·ing, pre·cludes 1. To make impossible, as by action taken in advance; prevent. See Synonyms at prevent. 2. the possibility of attention to the general. In the next excerpt Danny also focuses on one specific number, however, the structure that he attends to brings him closer to the solution.
Danny: Well, I'm trying to see what, what combinations, obviously
what combinations of numbers, I'm just trying to figure out
what multiples of hmm (pause). I'm trying to figure out the
pattern, repeating the same pattern by using the multiples
and by shifting them, to see if I could get the same
numbers highlighted. I could say, I could say that
multiples of 2, but it would highlight every even number
and I don't want every even number highlighted. I want, it
seems um every other even, every other even number
highlighted ... Perhaps I could say 2, shift 2 ...
Interviewer: Still gives you every even number ...
Danny: Every other number, yes. Maybe I'll go, uh go minus 2 ...
Interviewer: Good, multiples of 4, shift them by negative 2, you got the
same numbers. Can you explain why, please?
Danny: Can I explain why. (pause) Um, well I looked at, well I,
okay I looked at 240.
Interviewer: Okay 240 ...
Danny: I just, because I knew that 240 could be divided by 4, and
so I just minus 2 and then I could see, I could see the
same pattern, 1, 2, 3, 4, (pause) ... You look at 240, if
you add up 2 it brings you here ...
Interviewer: To 242 ...
Danny: You subtract 2, it brings us here, to 238 ...
In observing the pattern Danny describes the numbers highlighted as "every other even." However, he does not interpret immediately how "every other even" is related to the multiples of 4. He focuses instead on a "convenient" number 240, which he recognizes as being divisible DIVISIBLE. The susceptibility of being divided. 2. A contract cannot, in general, be divided in such a manner that an action may be brought, or a right accrue, on a part of it. 2 Penna. R. 454. by 4, and notices that the numbers highlighted close to 240, 242 and 238 are reached by adding or subtracting 2, respectively. This observation brings Danny to show multiples of 4 with shift of-2. This indeed leads to a desired result on the screen, which he does not completely understand at this point. Focusing on one specific number and its properties, rather than on a set of numbers and common properties of numbers in a set can be seen as reducing abstraction In object technology, determining the essential characteristics of an object. Abstraction is one of the basic principles of object-oriented design, which allows for creating user-defined data types, known as objects. See object-oriented programming and encapsulation. 1. (Hazzan haz·zan n. Variant of chazan. , 1998), a coping strategy students employ when they are presented with a problem that demands a level of abstraction The level of complexity by which a system is viewed. The higher the level, the less detail. The lower the level, the more detail. The highest level of abstraction is the single system itself. that is higher than the level on which students operate. Hazzan described attention to an element of the set, rather than a whole set, as one of the strategies students utilize as means to cope with the complexity of a situation. Focus on a specific relationship The problem presented to the participants involves three specific numbers: grid width of 9, show multiples of 4, and shift by 2. There are many possible ways to connect these numbers to each other in an arithmetic relationship. However, most of these relationships are irrelevant to the problem. Among popular initial attempts was to switch inputs and show multiples of 2 with a shift of 4. Though only two participants provided an explicit justification for this choice, focusing on 6 as the first number to be shown with both multiples of 4 shifted by 2 and multiples of 2 shifted by 4, we believe participants who didn't did·n't Contraction of did not. didn't did not didn't do explain this explicitly were guided by a similar relationship. A focus on 6 could also cause Aimee Aimee, or Aimée, is a female given name and a version of Amy. Both names come from Aimée, which means beloved/loved in French (for a female), from Old French amede, from Latin amāta, feminine singular past participle of and Blake to show multiples of 6 with shift of 0; as Blake explained: "Well in theory multiples of 4 plus 2, is really just multiples of 6, multiples of 4, put one in every 4th spot, plus 2 would be in every 6th spot." Furthermore, attention to number 6 may have led Mary Mary, the mother of Jesus Mary, in the Bible, mother of Jesus. Christian tradition reckons her the principal saint, naming her variously the Blessed Virgin Mary, Our Lady, and Mother of God (Gr., theotokos). Her name is the Hebrew Miriam. to show multiples of 3 with a shift of 3 and John to show multiples of 5 with a shift of 1. We provide here several additional examples in which participants attend to a numerical numerical expressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive. numerical nomenclature a numerical code is used to indicate the words, or other alphabetical signals, intended. relationship that is marginal to the problem at hand. * In a previous excerpt we mentioned Alice, who considered multiples of 4 shifted by 13. This was shortly after her success with shifting multiples of 4 by 6, where 6 is the sum of 2 and 4, two of the original inputs. She explained that 13 was the sum of 9 and 4, two other original inputs. It appears that Alice was attending to an arithmetic relationship of the inputs and not the structure of the numbers displayed. * After several unsuccessful attempts Blake declares being stuck. To help him, the interviewer shows that the desired result can be achieved by showing multiples of 4 shifted by 6. The interviewer further asks that Blake consider other possible inputs to shift. Consequently, Blake changes the shift to 12. At the request of the interviewer to explain this conjecture, he replies that "12 is 6 times 2," as if something obvious should result from this relationship. * After several unsuccessful attempts to focus on one number (as described in a previous section), Kori shows multiples of 4 shifted by 6. The following is her explanation: "Does it have anything to do with 6 being 2x3, and it was 2 and you multiply mul·ti·ply v. 1. To increase the amount, number, or degree of. 2. To breed or propagate. that by 3, you get 6 and 3 is a factor of 9, the grid width ..." We consider the above as examples of what Hewitt (1994) refers to as "train spotting." He uses this term to refer to situations in which students focus on surface patterns rather than on the deeper structural relationship that supports the pattern. These examples of pattern spotting can also be seen as a different kind of reducing abstraction. Rather than attend to a relationship that holds for all possible inputs, that is, a set of relationships, students focus on a relationship that is specific to one case, and in so doing, reduce the situation to a less complex one. Chain of conjecturing In previous sections we described factors that may influence the making of a conjecture. In this section we follow the testing of a conjecture. There are two obvious outcomes from such testing: the conjecture can be either verified or falsified. Both verification and falsification falsification /fal·si·fi·ca·tion/ (fawl?si-fi-ka´shun) lying. retrospective falsification unconscious distortion of past experiences to conform to present emotional needs. may either break a chain of conjecturing or continue this chain. In what follows we illustrate what we mean by breaking or continuing the chain of conjecturing with several examples. Let us consider first the falsified conjectures in the following two episodes: * Mary showed multiples of 3 with a shift of 3. Having found that "this doesn't work, let's try something else," she showed multiples of 8 shifted by 2 and then multiples of 5 shifted by 2. "I don't think anything else will work" was her conclusion following several additional experiments. * Katie Katie may refer to: In sports:
A superficial superficial /su·per·fi·cial/ (-fish´al) pertaining to or situated near the surface. su·per·fi·cial adj. 1. Of, affecting, or being on or near the surface. 2. look at these two participants reveals that they are experimenting with different inputs, one having greater success than the other has. However, we see that Mary's chain of conjectures breaks after each test. That is to say, she does not use what can be learned from the output (or at least what she learns is not obvious and is not shared with the interviewer) to inform her next input. That is why we refer to her chain of conjectures as broken or disconnected. Katie, on the other hand, reflects on the output of her conjecture and uses it to make the next conjecture. When there are too many numbers shown, she chooses an input that will lead to fewer numbers shown and vice versa VICE VERSA. On the contrary; on opposite sides. . (It is interesting to note that her line of thought is similar to Amber's. However, Katie runs the testing of her conjectures on a computer, while for Amber it is a thought experiment.) Now let us turn to considering a positive outcome, a verified conjecture. The input of -2 to the shift showed desirable results for several students. Had it happened outside of the interview setting, we believe, some participants would have stopped their work on the problem, having answered the question. However, whenever a participant did not continue the experimentation on her/his own, the interviewer prompted further, often asking to look for additional inputs that result in the same number display. Having faced a success with -2, Amber (excerpt shown above) turned to consider other even numbers. Celia used -6 as her next input. Jessica attempted to generalize, believing that if positive shift works, the same negative shift will work. Andrew, having verified the conjecture of shifting -2, concluded that "any multiple of 4 plus 2, either positive or negative" would give the desired result. These responses show different levels of generality gen·er·al·i·ty n. pl. gen·er·al·i·ties 1. The state or quality of being general. 2. An observation or principle having general application; a generalization. 3. , moving from generalizing for one specific input to drawing a general pattern of workable results. However, rather than discussing different levels of sophistication so·phis·ti·cate v. so·phis·ti·cat·ed, so·phis·ti·cat·ing, so·phis·ti·cates v.tr. 1. To cause to become less natural, especially to make less naive and more worldly. 2. they represent, we wish to show how a verified conjecture may continue the cycle of conjecturing, rather than end or break it. This is in accord with looking back, Polya's step #4 in problem solving (understand the problem, design a plan, carry out the plan and look back), that includes, among other heuristics heu·ris·tic adj. 1. Of or relating to a usually speculative formulation serving as a guide in the investigation or solution of a problem: , searching for alternative solutions or solution approaches, and generalizing the results (Polya, 1957/1988). Teacher's role It is generally believed that conjecturing is a worthwhile mathematical activity in which teachers are urged to adopt a supportive and non-judgmental approach. Computerized environments change the nature and possibilities of conjecturing as well as the role of the teacher. Traditionally, the role of the teacher is described as supporting students in testing their conjectures (NCTM, 2000). However, in microworlds such as Number Worlds, the computer instantly does the job of verifying or falsifying. So what becomes the role of the teacher? We believe that the main role of the teacher is in helping students succeed in not breaking the chain of conjecturing, that is, in helping them interpret the computer output and use it as feedback in continued conjecturing. This is true both for conjectures that have been falsified as well as for those that have been verified. Taking this approach shifts the focus from static conjectures to the dynamic process of conjecturing. It emphasizes the process of conjecturing as being located within a larger motivational and epistemological e·pis·te·mol·o·gy n. The branch of philosophy that studies the nature of knowledge, its presuppositions and foundations, and its extent and validity. [Greek epist context. Summary and Conclusion In the mathematics education literature, students' conjecturing is traditionally discussed in the context of proof, justification and reasoning. Edwards (1997) describes conjecturing as belonging to the territory before proof that is, a range of activities in which students may engage prior to creating a proof. As a complement to this view, our study advances the understanding of conjecturing on two counts: First, by "zooming out" we situate conjecturing in a broader set of experimental approaches that students adopt when faced with a problem. Second, by "zooming in" we refine conceptions of what is involved in the activity of conjecturing. Specifically, we describe the factors that influence how a conjecture is generated and what happens after it is tested. Attending to the mathematics of the particular situation described here, we note that only a few participants (6 out of 20) decided to keep the input to show multiples of constant in their initial experiments. Others focused on particular numbers or on particular relationships, without attempting to capture the structure. This may appear surprising, given that all the participants had no difficulty in recognizing a visual display of disconnected diagonals as a display of multiples (Sinclair, Zazkis & Liljedahl, 2003). It could be that the participants' awareness of the distribution of multiples, that is, every nth number is a multiple of n, is more easily recognized in a display than utilized in a problem situation. Our further contribution is in articulating the teacher's role in supporting students' conjecturing. When conjectures are tested with the help of a computer, the teacher's role shifts from helping students test their conjectures to helping students interpret both verified and falsified conjectures and continue the chain of conjecturing, rather than ending or breaking it. Therefore it is beneficial for preservice teachers to develop awareness of their own conjecturing in order to guide the conjecturing process of their students. We suggest that teachers can start gaining this awareness by attending more explicitly to the kinds of experimental approaches involved in the process of conjecturing. References Borwein, J., & Bailey, D. (2004). Experimental mathematics: Computational paths to discovery. Boston, MA: AK Peters. Cuoco, A., & Goldenberg, P. (1996). A role for technology in mathematics education. Journal of Education, 178(2), 15-32. Edwards, L. (1997). Exploring the territory before proof: Students' generalizations in a computer microworld for transformation geometry In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein, who pioneered this point of view, was himself interested in mathematical education. . International Journal of Computers in Mathematical Learning, 2(3), 187-215 Goldenberg, P. (1989). Seeing beauty in mathematics: Using fractal geometry fractal geometry, branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry. to build a spirit of mathematical inquiry. Journal of Mathematical Behavior, 8, 169-204. Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction CONTRADICTION. The incompatibility, contrariety, and evident opposition of two ideas, which are the subject of one and the same proposition. 2. In general, when a party accused of a crime contradicts himself, it is presumed he does so because he is guilty for and uncertainty in promoting the need to prove in dynamic geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts. environments. Educational Studies in Mathematics, 44 (1-2), 127-150. Hazzan, O. (1999). Reducing abstraction level See level of abstraction. when learning abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, concepts. Educational Studies in Mathematics, 40(1), 71-90. Hewitt, D. (1994). Train spotter's paradise. In M. Selinger (Ed). Teaching mathematics (47-51). London: Routledge. Hewitt, D. (2001). Arbitrary and necessary: Part 2. Assisting memory. For the Learning of Mathematics, 21(1), 44-51. Mason, J., Burton, L., & Stacey, K. London: (1982). Thinking mathematically. Addison Addison, village (1990 pop. 32,058), Du Page co., NE Ill.; inc. 1884. An industrial suburb of Chicago, it manufactures machinery and plastic items. Wesley. National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. . (2000). Principles and standards for school mathematics Principles and Standards for School Mathematics was a document produced by the National Council of Teachers of Mathematics [1] in 2000 to set forth a national vision for precollege mathematics education in the US and Canada. . Reston, VA: Author. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings. Dordrecht, The Netherlands: Dluwer Academic Publishers. Polya, G. (1988). How to solve it: A new aspect of mathematical method (2nd Edition). Princeton, NJ: Princeton University Princeton University, at Princeton, N.J.; coeducational; chartered 1746, opened 1747, rechartered 1748, called the College of New Jersey until 1896. Schools and Research Facilities Press. (Original work published 1957). Sfard, A. (1991). The case of the vanishing tuples. For the Learning of Mathematics, 11(1). 19-25. Sinclair, Zazkis & Liljedahl, (2003). Number Worlds: Visual and experimental access to elementary number theory concepts. International Journal of Computers in Mathematical Learning, 8(3), 235-263. Zazkis, R., (2001). From arithmetic to algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as via "big" numbers. Proceedings of the International Congress of Mathematics Education Study on Algebra. Melbourne, Australia. Zazkis, R. & Liljedahl, P. (2002). Arithmetic sequence as a bridge among conceptual fields. Canadian Canadian (kənā`dēən), river, 906 mi (1,458 km) long, rising in NE New Mexico. and flowing E across N Texas and central Oklahoma into the Arkansas River in E Oklahoma. Journal of Science, Mathematics and Technology Education, 2(1), 91-118. Rina Zazkis, Simon Fraser University Simon Fraser University, main campus at Burnaby, British Columbia, Canada; provincially supported; coeducational; chartered 1963, opened 1965. The Harbour Centre campus in downtown Vancouver opened in 1989. Nathalie Sinclair, Michigan State University Michigan State University, at East Lansing; land-grant and state supported; coeducational; chartered 1855. It opened in 1857 as Michigan Agricultural College, the first state agricultural college. Peter Liljedahl, Simon Fraser University |
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