Purposeful contexts for formal notation in a spreadsheet environment

JOURNAL OF MATHEMATICAL BEHAVIOR 15, 405-422 (1996)

Purposeful Contexts for Formal Notation in a Spreadsheet Environment

JANET AINLEY

University of Warwick

This article addresses the early stages of children’s introduction to the use of variables in

formal algebraic notation. We conjecture that some of the difficulties encountered by

children in this area may be accentuated by their lack of appreciation of the purpose or

power of formal notation. A teaching approach is described that aims to situate the use of

formal notation in meaningful contexts. A study of a teaching sequence based on children

working with this approach, using graphical feedback in problem solutions, is presented.

BACKGROUND

In a recent survey of the learning and teaching of school algebra, Kieran (1992) cited a number of research findings that indicate the relative success of computer- based environments in developing children’s understanding of variables in the early stages of learning algebra. Kieran attributed this success largely to the procedural nature of the programming involved. The use of variables in Logo is mentioned particularly as being accessible because it lends itself to procedural interpretations. Kieran also commented on the fact that although there has been a great deal of research into children’s learning of algebra, there has been little research into the teaching of algebra or the content and presentation of what is taught.

This article reports on research that involves an innovative approach to the introduction of the use of variables to primary school children. This approach is based on the conjecture that the lack of any sense ofpurpose for the use of formal algebraic notation in traditional approaches to beginning school algebra may contribute to children’s difficulties in accepting formal notation. In contrast to many traditional approaches, activities based around working with a computer often involve pupils in using variables, for example within Logo programming, in order to achieve particular effects. Here the algebraic notation is a means, rather than an end in itself. The fact that children meet variables in a context where there is a clear purpose for their use may suggest an additional explanation for the relative success of children working in computer-based environments.

Correspondence should be sent to Janet Ainley, Mathematics Education Research Centre, Insti- tute of Education, University of Warwick, Coventry CV4 7AL, United Kingdom.

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A further aspect of computer-based environments is that the software requires the user to communicate by means of a restricted code governed by formal conventions. Unlike a human colleague or teacher, the computer cannot apply common sense or interpret half-formed ideas. This means that formal notation seems natural and purposeful in a computer environment in a way it does not in human communication. Thus we might see the use of formal notation in a computer-based environment as potentially purposeful at two levels: There is a purpose in using it to communicate with the computer, and this is itself within’the context of producing a further end result.

APPROACHES TO CONTEXTUALIZING ALGEBRAIC NOTATION

The idea of contextualizing formal notation is clearly not a new one. Word problems have been widely used to offer a way both of giving meaning to algebraic expressions and of linking work in algebra to children’s experiences of arithmetic problems. However there is considerable evidence that representing word problems as formal equations presents major difficulties for pupils (Kieran, 1992). In particular, difficulties arise from what Chaiklin (1989) referred to as a direct translation approach, in which pupils try to form algebraic expressions by modeling the structure of the verbal statement, rather than that of the underlying mathematics.

A further difficulty with this approach is that such word problems generally have a single solution, which may be found by a number of difftcult methods, some of which will be purely arithmetic. Describing the problem situation in an algebraic form may be high on the teacher’s agenda, but it may seem an artificial complication for pupils who can reach an answer using a different method.

investigations offer another approach to introducing formal algebraic notation in meaningful contexts. This approach is widely used in assessed coursework for the General Certificate of Secondary Education in the United Kingdom. Typ- ically in such an activity the child might be required to explore a number pattern arising from a practical situation, and then he or she might be asked to find the hundredth number in the pattern, or for a method for finding any term in the sequence. The teacher’s aim is to encourage the child to generalize the pattern in the form of an algebraic expression.

This approach has been characterized by Hewitt (1992) as train spotting, because the learner’s attention is generally focused on pattern spotting rather than on the situation from which the investigation arose. Even when the initial situation was one in which the mathematical structure was accessible, such as a pattern built up from match sticks, this is overlooked in the attempt to find patterns in the resulting tables of numbers. From the child’s point of view, it is difficult to see any purpose in formalizing the pattern in algebraic terms: A verbal

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description of the pattern, or a generic method for calculating values, may seem just as efficient for giving the solutions required.

AN ALTERNATIVE APPROACH TO FORMALIZING

The research described here took place within the Primary Laptop Project, in which a computer-rich environment has been created. One focus of research in the project has been children’s use of spreadsheets as a mathematical tool. Early studies indicate that the children’s ability to interpret and understand graphs has been enhanced through working in a spreadsheet environment (Ainley, 1994, 1995). In order to exploit this potential, we have developed a teaching approach (illustrated crudely in Figure 1) that we call active graphing (Ainley & Pratt, 1994a). Children are encouraged to enter data they collect in experimental activities directly into a spreadsheet. They then graph these data regularly during the course of the experiment, enabling the graph to be used as an analytical tool that helps them decide which data they need to collect next.

This means that the physical experiment, the tabulated data, and the graph are brought into close proximity. Research evidence from data-logging projects (e.g., Mokross & Tinker, 1987) supports our conjecture that this proximity is important in supporting children’s understanding of the conventions of graphing and their ability to interpret complex graphical representations by relating them to the activities from which they arise (Pratt, 1994).

Because the spreadsheet is an environment in which an algebralike notation is used, we were interested to explore whether an active graphing approach could be used to introduce children to the power of generalizing through formal algebraic notation. In order to do this, we selected activities that lent themselves to this approach. There needed to be a practical element, so that children could

I collect initial data t

study graph and make refine conjectures

l

. decide what further data is needed

when you are ready!

Figure 1. The active graphing process

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begin by collecting data, but also an element in which the underlying mathematical structure was accessible to the children. Two other key features of the activities were that the outcome was not obvious, so that there was some point in using the active graphing approach, and that the practical activity was in some way problematic, so that children would be encouraged to look for short cuts.

With these criteria in mind we selected a number of activities, several of which involved optimization. We have reported elsewhere (Ainley & Pratt, 1994b) on the use of one of these as a whole-class activity. This gave us some insights into the stages children went through in working on the activity, and the situations that prompted the need for formal notation. This article focuses mainly on one pair of children working on a second activity, known as The Sheep Pen (shown in Figure 2).

Methodology In this stage of the Primary Laptop Project our research is essentially exploratory, rather than addressing clearly focused research questions. We are interested in exploring the range of mathematical activities that are possible for children who have continuous and immediate access to computers, and identifying areas for more focused research in the future.

The case study material used in this article was collected in a research setting removed from the classroom. In all, eight pairs of children (chosen by the researchers) worked on the activity with one of the researchers acting as “teacher,” introducing the activity, responding to the children’s questions, and occasionally intervening. The sessions were recorded on videotape, with the second researcher also taking field notes.

Jordan and Stellios were both 11 years old and in their final year at primary school. They were described by their class teacher as being of average ability, but

A farmr has 3Cxt-1 of flexible fencing. Shewantstormke a rectangularpen for her sheep against a stone wall.

k.. ,.. ,. >. : .:. ..; ,,.,,>,;, .,.3 ‘M~W~~~ sv<w .%A~ .Y.+=v %WY,X

What lengthandwidthshould shemkeit to enclose the largest area?

What if she had a different length?

Figure 2. The sheep pen activity.


neither of them was particularly highly motivated in mathematics. They had not been introduced to formal algebraic notation, but they were familiar with using a spreadsheet and had experience of an active graphing approach in the context of experimental work.

Working Through the Active Graphing Process Jordan and Stellios began by working practically on the activity. They were provided with an art straw cut to 30 cm long to model the fencing, and had a ruler available (which also happened to be 30 cm long). They bent the straw, measured the length and width of the pen they had made, and set up columns on the spreadsheet to record their results. From previous experience, they knew that they could use a formula to calculate the area of the pen, and fill down the column to replicate the formula. Because the focus of the activity was not on understanding the calculation of area, the researcher gave whatever help they needed to get this formula working correctly.

When they had made several model sheep pens by folding the straw and measuring the length and width and entered the data in the spreadsheet, the researcher intervened to encourage them to create a graph of their results, shown in Figure 3. At this point some of their data were inaccurate, but they did not immediately pick this up from the graph, because they had no particular expecta- tions of the shape the graph would take.

Jordan was able to discuss the meaning of the graph, but at this stage his attention was on particular points, rather than on the relationship between width and area.

RES: What seems to be happening there, as far as you can tell? JOR: Well, the best width is 8, and it goes up its the biggest. The worst is urn that

one (pointing at width 2). RES: Might it be possible to get any that are even worse than, what width was that? 2? STEL: How would you put in a half. RES: If we did a half, a width of a half, how would that fit in with the crosses we’ve

got at the moment? JOR: It wouldn’t appear on that graph because the lowest we’ve got is 2 cm (pointing

to the horizontal axis) so it would move on a bit (waving his hand to the left) and have a half.

However, it is clear from the boys’ responses to further questions that they were aware of the overall shape and pattern of the graph.

STEL: If I put eight and a half, where would that be? How would we write that? RES: Where do you think 8.5 would appear as a cross?

(Stellios points between 8 and 9 for the width, and at about 100 for area. They put in 8.5 as the width, and Jordan bends the straw and measures the length as 14. The area appears on the spreadsheet.)

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STEL: Highest! That’s the best so far! (Jordan makes a chart again to check the position. The length is actually measured incorrectly, so this point looks higher than 8 or 9.)

JOR: There it is (pointing to the graph). STEL: (pointing lower) I thought it would come around here.

The boys were confident to make predictions based on the graph, but they had not yet seen the shape of the graph clearly enough to realize that some of their measured data were inaccurate. (In observing other pairs working on similar activities we had noticed that irregularities in the graph often provided feedback that stimulated them to question their results. Typically they then either re- measured, or changed to calculating the length of the sheep pen for a given width.)

The boys’ next choice of width provided the stimulus for them to calculate the corresponding length, rather than measuring. It was quite awkward for them to bend the straw accurately for a very small width, and the numbers involved made the calculation relatively simple.

STEL: Try a width of point 5. JOR: What’s the length? STEL: Oh er 19, 29 RES: How did you work that one out Stellios, because you didn’t measure that one did

you? STEL: If the ruler’s 30, half and half is one and the rest is 20, no 29.

Once Stellios had described his method, the boys used it to put in this new value, and then drew another graph. They then decided, without prompting, to check the other values they had already entered using Stellios’ method. Thus the method Stellios had devised initially for finding a single value developed into a generic method they could use repeatedly.

At this point the researcher intervened to suggest that the boys might teach the computer their method for calculating further data. The metaphor of teaching the computer was one familiar to the children from their work with Logo.

RES: What you are trying to do is to tell the computer how to work the length out, given some width (pointing to cell Bll in the width column). So if you knew what that width was, you’re trying to work that length out (pointing to cell Al 1, in the length column).

JOR: You have to add these together (pointing vaguely at the length and width column). double it (pointing to the width).

STEL: How do you double it? JOR: and then you work out the length. STEL: zero point five add zero point five or something JOR: yeah but they don’t know (pointing at width cell).

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JOR:

STEL: JOR: STEL: JOR: STEL: JOR:

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IknowBeleven(typing),B ll,Bll,. .rightB ll,add,. .Blladd,ohno, B 11 times 2. oh yeah times 2 so then that doubles it, and add A 11 Bll times2add.. . add Al 1 equals Cl 1 No we need to . if there’s 30 in the ruler right, it’s all doubled though, we need to tell it how to work out what’s left.

The boys’ initial attempts to formalize their method show a number of significant features. Jordan has a clear picture of the calculation he wants to express, but has to overcome two hurdles in order to formalize it. The first is to express “double it,” which he quickly resolves as “times 2.” The second is more difficult. Having doubled the width, he then needs to find a way to express “what is left” from the original 30 cm. In working on this, the boys quite confidently use B 11 as a placeholder for a width they do not yet know. This step in formalizing does not seem to present an obstacle for them, although later, when they try to resolve the problem of how to find “what is left,” Jordan reverts to a particular example, as seen in the next extract. His use of the cell reference as a placeholder is not completely secure: He needs the support of an example when the problem becomes more complex.

The boys continued to work on their problem for several more minutes, occasionally touching the keyboard, but mainly trying out ideas verbally. At one point, they deleted the formula they had typed, and the researcher took the opportunity to ask them to recap what they had done.

JOR: STEL: JOR: RES: JOR:

RES:

JOR: STEL: JOR:

RES: JOR:

So far we’ve got, from here we’ve got Bl 1, anything that’s in Bll. Times it by 2. times it by 2 so it doubles it

. . OK We need to tell it like, we want to tell that there’s 30 over there, if we times, say it was 5, times by 2 it becomes 10, and what, and tell it to know how much is left on the ruler. Right. How do you calculate what’s left? What do you do when you do it in your head? Well if it was, if it was . What’s left . . . is it that little r thing? Is it remainder? If it was, if it was 7, you double the 7 to 14 it would go in there but there’s 16 left

What have you done to work that 16 out? I know that 14 add 16 is 30.

Here although Jordan reverts to an example when he cannot resolve the problem of finding what is left, his grasp on B 11 as a placeholder has changed.


He spontaneously talks about anyfhing r/rat is in Bll, indicating he has some understanding of the cell reference as a variable. It is interesting that each time he goes to an example he chooses different values to work with. This suggests that he has moved a long way toward generalizing the method. When he resorts to working with a particular value, it is as a generic example rather than as a particular case. He uses a particular value to illustrate or specialize his method.

Stellios’ interjection about remainder in this extract superficially seems con- fused, though as shown in later discussion it gives a significant clue to the boys’ thinking.

About 10 minutes later, they decided they needed to include 30 in their formula. They typed

=30 BI 1*2

They seemed to have a sense here that they must start with the length of the straw, but their verbalization was “take it away from,” and they could not see which operation to use. They quickly deleted this formula and typed

=Bll*2-30

STEL: You can’t take 30 from . . . urn JOR: times it by 2 take it from 30

STEL: times it by 2 and take it from 30 They try putting in 13 for the width and get length -4 and area -52. JOR: it’s probably 52

STEL: the minus, shouldn’t have put the minus in JOR: I don’t know. JOR: Blltimesitby2takeitfrom30.. . but this looks like take away 30, and we

don’t . . It should have been 4, so it’s nearly right.

At this point, the boys had been working on the problem of teaching the computer their method for around 30 minutes. It is tempting to interpret their position at this point as failure to move from the method they had verbalized to a formal algebraic expression. However, from the language that Jordan uses it would seem that he has accepted the cell reference as a variable he can operate on. The researcher felt that Jordan’s difficulty lay in attempting to translate their verbal formulation of the method into the formal notation of the spreadsheet, when in fact the spreadsheet could not “understand” the way they had expressed the arithmetic.

Their verbal formulation “take it from 30” was quite adequate for most pur- poses and it communicated their thinking clearly. It followed closely the physical process they had gone through, choosing the width and bending the straw this amount at both ends (“Bll times 27, then measuring the length of straw left between the two folds (“take it from 30”). The researcher decided to intervene,

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offering them a slightly different physical model with the aim of redirecting their attention from the verbal formulation and back to the physical situation. The results were dramatic.

RES: Let’s think of it in a different way Here’s our length of fencing, which is 30 (holding up straw). Let’s imagine cutting off our two widths. So we’re starting with the 30 and instead of folding, let’s cut them off.

JOR: If we start with 30, take away B 11 times 2 Jordan typed in the correct formula (=30 - B11*2), filled down the column, and they began to enter more values for the width. JOR: we virtually did that, but it was the other way round.

The boys then worked excitedly, entering values to try to find the maximum area, and using decimals to home in on where they thought it would be. They soon had too many crosses on their graph to be able to use it effectively to read off the maximum value (see Figure 4).

They then concentrated on looking at values on the spreadsheet, and came across problems in reading numbers containing different numbers of places of decimals, and they sorted the data on the area. Stellios was puzzled about the computer showing the area of 112.5 as bigger than 112.48. He seemed to think the computer has used a rule that was not right: He said, “It’s bigger to the computer.” Jordan explained that it is bigger because .5 is a half, and .48 is less than a half. Stellios seemed to accept this by imagining both figures as two places of decimals: “So it’s like adding ‘oh’ on to there [to make 501.”

120 ..................................

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100 ....... .............. .x.:. ................ .x_ ............... ...... ........................

90 ............. ............... ..............

A 8o ... ........... ..... ........................................

R 7o .............. )( __ ................ ................. ........................ ............

E 60 ._._. ............... .__ ................ .......................... ...................... ......

A 50 ...... .x ._._ ................... .................... x.. ..............................

40 ............... ...... .............

0 4 8 12 16 20

WIDTH

Figure 4. Graph drawn from calculated data, showing “homing in” around 7.5.


Figure 5. A handwritten spreadsheet.

Finally they realized that a width of 7.5 cm gave the maximum area. In another session on the following day they worked on the more general problem of finding the largest pen for any length of fencing. Initially they repeated the active graphing process taking 20 m as the length of fencing. After some difficulties, they adapted their formula to calculate the length for any width, and enjoyed putting in more and more values for the width to produce graphs showing smooth curves.

When they had found the dimensions of the pen with the maximum area starting with 20 m of fencing, the researcher focused their attention back to the general problem, asking if they could give any instructions to the farmer. After some discussion, Jordan wrote, “To get the highest area the width of your pen must be a quarter of the total length of fencing, and the length of your pen must be half of the total length.” With some prompting from the researcher, they managed to rewrite this instruction (on paper) as spreadsheet formulae (see Figure 5).

DISCUSSION

In analyzing the work of Jordan and Stellios, a number of factors seems to contribute to their success in formalizing. These are discussed by looking back at the case study in terms of three layers of detail: the stages in the activity; the

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changing use of the cell reference; and the links between practical activity, verbalization, and interaction with the spreadsheet. In looking at these layers, experiences from observations of other pairs working on similar tasks has also been invoked.

Stages in the Activity Although children tackled the activities in different ways, a number of general stages can be identified in their progress through the active graphing approach. These are summarized in Figure 6.

The problem was initially posed using the practical materials, and almost without exception children began by collecting measured data. The move to calculating values happened spontaneously (i.e., without teacher intervention) for most pairs, either in response to feedback from the graph showing that some points were obviously wrong, or when looking at extreme values, as for Jordan and Stellios. A few children quickly became exasperated with the practical

practical activity

I

often triggered by need to check points, or by looking at extreme cases

mental method

1 often triggered by checking existing data

generic method

I intervention to suggest ‘teaching the computer’

verbalization

I)71

spreadsheet formula

1 generating data, graphing and interpreting solution

Figure 6. Stages in the activity.


activity, and looked for other methods; others worked more methodically and produced very clear graphs from measured data.

Once they had a method for calculating data, most pairs realized that they could use this to check the data they already had. For Jordan and Stellios, the stimulus to do this came from feedback from the graph, when the mixture of measured and calculated data showed inconsistencies more clearly. When the method was used in this way for calculating several values, it could be regarded as a generic method, and this in turn became a signal to the researchers to intervene to move the activity on, by suggesting that the children could teach their method to the computer.

A few children ignored this suggestion. One or two pairs were able to move very quickly to producing a formula, but for most pairs, like Jordan and Stellios, this proved to be the most extended and demanding stage of the activity. As they discussed the task and tried out ideas on the computer they moved backward and forward between particular examples (often linked through gestures to the practical situation), increasingly precise verbalizations of their method, and attempts at producing a formula. During this period, their grasp of the generic method became more secure-Jordan particularly used a range of examples with ease- and their ways of expressing the method verbally became more fluent.

For some pairs of children, it seemed that a further stage occurred here: the move from a generic to a generalized method. The distinction here is subtle, and as difficult to detect with confidence as it is to describe. We might describe having generalized the method as being able to see it as a relationship between the variables (length and width), rather than as an operation performed on particular values or cells. The only clues available to the researchers about whether or not children had reached this stage was in their verbalizations. For example, some children verbalized their methods in terms of the width and the length, rather than in terms of particular values, or cell references. So Stellios’ method would become “To find the length, double the width and take it from 30.” We have no evidence of Jordan and Stellios making such statements, although they successfully moved on to create the formula they wanted. The possible signifi- cance of the move to a generalized method is discussed later.

Jordan and Stellios were typical of many children in that they had a long period with apparently little progress before they produced a formula, and then became very animated in the last stage of the activity, generating a lot of data, drawing graphs, and exploring solutions. For them, this was the most significant part of the activity.

Changing Use of the Cell Reference Jordan’s increasingly sophisticated use of the cell reference reflects stages typical of many of the children observed. At first, he used it just as an alternative name for the number it contains, when discussing aspects of the spreadsheet layout. Later, Jordan used it as a placeholder for a potential number soon to be realized

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(“Yeah but they don’t know . . [pointing at width cell] I know B eleven [typing] B 11.“). Finally, he seems to be using the cell reference as a placeholder for a range of numbers that could be put into it (“So far we’ve got, from here we’ve got Bl 1, anything that’s in Bll.“). Jordan’s hand movements suggested that he may have been visualizing the column of figures below the particular cell, into which they will later be able to enter numbers, once they have created the formula.

One feature of these children’s previous experience with spreadsheets is that they had been introduced to the use of formulae in contexts where they had always immediately replicated them to generate columns of data. Indeed, some of the pairs we observed clearly saw these as two parts of the same process, trying to include the information about which cells they wanted to copy to in the formula itself. This may have provided a powerful mental image for the location of a range of values, represented by a single cell reference. Because of this, it may be debatable whether Jordan is really using the cell reference as a variable, as opposed to a collection of particular values. Rather than trying to unpick this further, we prefer to consider the image of an empty column as a powerful image of the ambiguity between the particular and the general.

In the final stage of the activity, Jordan used the cell reference in his handwritten spreadsheet, despite the fact that the length of the fencing was given in the adjoining cell. By this point, he seems confident to move from the specific to the general; In writing the specific cell reference he assumes the generality of the column of spaces below, where it is this reference, rather than the specific value, that is to be used.

Tall (1992) referred to a formal algebraic expression of a relationship as a template, a potential arithmetic relationship waiting to be realized. Some children may only be prepared to accept the use of a symbol as a placeholder within the template if that potential can be immediately realized (i.e., it can be immediately turned into a number). Later, children may accept a greater distance between the use of symbolization and its realization as a number. Such children are further on the way toward accepting that the symbolic expression is itself something that can be manipulated and used, as if realizing the potential could be delayed indefinitely.

It is this last step we equate with children generalizing their method for calculating data. When working with a spreadsheet, it is difficult to identify those children who have reached this final level of sophistication in their thinking, because those with more limited views of the nature of the cell reference may also be able to successfully create a formula to model their rule. We conjecture that the extent to which children are able to express a verbal generalization of the rule they are trying to formalize may give some indication of whether or not they have taken this final step in their thinking. Much of Jordan and Stellios’ discussion of the problem focuses on creating a formula: They repeatedly use the cell


reference, and so it is often unclear how far they have moved toward such a generalization.

Interaction With the Spreadsheet It was Stellios’ apparently random remark “Is it that little r thing? Is it remainder?’ that provided a clue as to how some of the interactions between the boys’ verbalizations of their method of calculation, and their attempts to create a formula, might be interpreted.

Stellios’ remark is puzzling, and indeed we watched the tape several times before we noticed what he was saying. On reflection, he seems to be making a link between the phrase “what’s left over” and memories of division problems, where hehas learned to record the remainder with a little r (e.g., 25 + 3 = 8 r 1). He seems to be taking a direct translation approach (Chaiklin, 1989), focusing entirely on the words that are being used to describe the method, and disregard- ing the situation from which the method arose.

With this insight, it is possible to look at stages in the boys’ verbalization of their method to see the extent to which they are supported or mislead by using direct translation. When they first began trying to create the formula, they slipped quickly between three different verbalizations of the first step in their method, doubling the width. Jordan verbalized this first as “You have to add these together” (pointing vaguely at the length and width columns), then realized that he only wanted the width, and changed to saying “double it” (pointing to the width). Stellios’ question “How do you double it?’ seems to prompt him to a third version:

JOR: Bll add, oh no, Bll times 2. STEL: oh yeah times 2 JOR: so then that doubles it

In fact both boys continued to use both addition and multiplication when working out particular examples throughout the rest of activity. In this case, they were able to slip between different verbalizations and ways of seeing a simple calculation.

The aspect of the boys’ method that caused them most difficulty was the final step of finding what was left. Their final verbalization was “times it by 2 take it from 30,” and the boys’ attention became fixed on translating this directly into a formula. This method seemed to arise from the physical actions the boys went through with the straw, choosing the width first, folding both ends of the straw, and then measuring what was left. Although there was nothing wrong with their approach as a mental method, it was not one that could be translated directly into the formal notation of the spreadsheet, or of standard algebra. If the boys had been working on the problem without a spreadsheet, the teacher would have had

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to make a decision about whether to accept this method or to push further to reformulate it into more standard mathematical language. In the context of the spreadsheet, it was important to move to a formula that the spreadsheet could accept, so that more data could be generated and graphs produced.

The fact that the verbal formulation the boys were trying to translate was one that they had produced themselves, rather than one given to them in a word problem, was significant in the approach that could be used to move them forward. The researcher was able to turn the boys’ attention back to the physical situation, and offer them a different way of seeing the process of making the sheep pen. The image of starting with the whole straw (fence) and cutting off the two widths had an almost immediate effect on Jordan’s ability to turn around the way he saw the situation and the arithmetic structure of the formula.

Approaches to the Spreadsheet Environment There are clearly different ways in which the same activity could have been pursued in a spreadsheet environment. For example, in some software it is possible to use a variable name for a whole column, so that instead of construct- ing their formula by using a particular cell reference, and them replicating this, the formula could have been built up using a reference to width. This seems to offer interesting possibilities for supporting the move from a generic to a generalized expression of the method. However, as this facility is not available in the software used in the project, this approach has not been explored.

It is very natural when working with a spreadsheet to point to cells and columns when discussing calculations: Indeed the software generally allows you to click on a cell in order to include it in a formula. Although this has been characterized as “point and grunt” technology, our experience is that this pointing supports rather than inhibits verbalization. In the style of Sutherland and Rojano (1993), we developed a teaching strategy of encouraging pointing, both with fingers and the mouse pointer, and using statements of the form “you want to add that to that and divide by 2.” However, in order to take full advantage of this approach, you need to develop different styles of using the spreadsheet to record the data. For example, in the sheep pen problem, the spreadsheet could be set up to show much more data than Jordan and Stellios chose to include.

A layout such as that shown in Figure 7 would be built up as you work on the problem: It would not be clear initially that the “double width” column is useful. Having all the information needed for the formula to calculate length already available on the spreadsheet may well make the creation of the formula easier, but arriving at this layout requires considerable experience of using a spreadsheet in order to know that it will be useful. It is not a starting point that seemed to come naturally to us or to the children. Imposing such a layout on the children from the start would, we felt, have detracted from their ownership of the

problem.


A B C D E

1 length of fence / width of pen j double width / length of pen i area of pen

2 _ __._______________._................ < .._..._._________............ 3.; . . . . . . . ..__________.................. ..__.________..._....................: . . . . . . . . . .!.oo.. 30 !

3 .._______.___.............. < . . . . . . . . . . . . ..__.___.................~ .._.___._.......... lo, ..________.......... Y”.. . . . . . . . . . . . . . 30 : 0 _

_ .._.._._____________.......... xl ..___.________._................ ~...........___ 4

5 30 j __________............ i . .._..______________................~ . . . . . . . . . . . . . . . 0..

0

6 30 j 0

Figure 7. An alternative spreadsheet.

Purposeful Activity In analyzing the tapes of pairs working on this activity, the perseverance the children showed in working toward a formalization of their rule is impressive. Jordan and Stellios spent about 30 minutes on this stage of the activity without noticeably losing motivation or moving off task for more than a short period, even when their attempts were apparently unsuccessful. Although they often talked in terms of operating on numbers or cell references, their hand movements indicated that their thinking was clearly grounded in images of folding and measuring the straws.

Even when situated in investigations or word problems, formalizing is often a separate process from the main activity that has been externally imposed by the teacher. In contrast, within active graphing activities, formalizing has a clear purpose: to generate more data. This larger quantity of data enables you to work on the problem, and the accuracy of these data can be seen from the feedback given by the graph. We conjecture that such activities give children a sense of the purpose and the power of formalizing. They realize that unlike their teacher, the spreadsheet simply will not be able to interpret nonformal rules, such as “take away from.” It is our belief that this experience of using formalizing contributes to children’s success in understanding variables. Like other computer-based environments, children’s thinking is supported by feedback given by the computer on their attempts to give a formalization. Further, there is an external referent, the physical situation in the case of the active graphing problems, or the functioning of the program in the case of programming. This broader context allows for alternative formulations to be developed, and so offers an escape route from the trap of direct translation from a single formulation.

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Documents

Purposeful contexts for formal notation in a spreadsheet environment