The equals sign as a focus for study

In this section I describe my research involvement with the VisualFractions

microworld. Trials were conducted with the software to test its general appro- priateness, usability and robustness prior to its commercial release byLogotron1_,

rather than explicitly to investigate conceptions of the equals sign. I first de- scribe the microworld and then summarise a trial with a pair of secondary pupils in which the symbol = unexpectedly arose as an issue of interest. The findings correlated with the literature on pupils’ conceptions of the equals sign (Chapter 2) and, I argue, demonstrate the value of technology as a window onto pupils’ thinking about formal notation. (For a full report of the study see Jones & Pratt, 2006).

VisualFractions presents learners with numbers (mostly fractions), operator signs (+,−,×,÷) and relation signs (=,<,>) as manipulable, on-screen ob- jects. Numbers (fractions) are available in a variety of forms including Arabic numerals, iconic graphics and number lines. The on-screen objects can be cre- ated, destroyed, altered and connected on the screen. For example, operator- and relation-objects can be connected to numbers and to other operators and relations. When connected correctly (according to the software’s functionality) operators return numerical results and relations return Boolean results. The equals sign, then, carries an “is the same as” meaning by design in the

VisualFractions microworld. A statement made by two pupils during a trial is shown in Figure 5.1. The operator sign is connected to the addends 13

6 and 8 12,

and returns a numerical result (the small fraction to the top-right of + reads

26

12). The equals sign is connected to two inputs, the result given by the operator

sign and the statement’s “answer” (14₆), and returns a Boolean result (“false” is written to the top-right of =). The “face” connected to the equals sign is a flag that conveys the Boolean result (thumbs-down means false).

The trial summarised here involved two girls aged 13 years, both of whom were confident and capable with mathematics and computing. The session lasted 80 minutes and was recorded using a microphone and screen-capture software. I was present throughout to introduce the microworld and task, and to offer support and prompt for verbal elaborations (e.g. “Why do you think that?”). The pupils were first introduced to some basic objects and allowed to experiment with making connections. They were then challenged to design an on-screen

Figure 5.1: An arithmetical statement inVisualFractions

activity sheet that might help primary children learn about fraction arithmetic.2

Audiovisual movie files were analysed using the qualitative analysis software package Qualrus3_{. A trace of the pupil’s activity was produced and used to}

develop a descriptive case study of key events. (A similar approach was used for the studies reported in this thesis and is fleshed out in Chapter 6.)

The two main findings were: (i) the children experienced much more difficulty when working with the equals sign than any other on-screen object; (ii) the pupils’ meanings for the equals sign were made apparent to myself as researcher by the functionality of the technology. These findings are elaborated below. The pupils readily and effortlessly got the hang of making connections between numbers and operators. When a disallowed connection is attempted the mi- croworld plays a flat note, which the girls correctly interpreted as an error noise. On the few occasions it sounded while working with number and opera- tor objects they simply stopped and tried an alternative connection. However, when it came to the = object they repeatedly attempted and returned to disal- lowed connections. For example, they tried several times to connect the equals sign to the addend immediately preceding it rather than to the + that outputs the result to be compared. They also repeatedly tried to connect the “face” (Figure 5.1) to the “answer”, rather than to the equals sign (which outputs the

2_{This task design draws directly on Harel and Papert’s (1991) seminal study into the}

pedagogic potential of the Logo programming language.

3_{Qualrus is a software package thatsupports qualitative analysis of audiovisual files.}

truth of the statement). These repeated attempts suggest the pupils attended to the left-to-right directionality of equality statements as would be expected from the literature (see Section 2.2). The pupils finally succeeded at connecting the equals sign when they switched their attention from a left-to-right sequence of symbols in favour of what might called a “data-flow” conception, as sup- ported by the microworld. This data-flow conception was evidenced by one of the pupils, Layla, referring to the statement as “a circuit”:

Layla: I think I connected the equals sign straight to the smiley face

instead of connecting it to the answer.

Carol: You’ve got the equals answer to the plus and then you’ve got it

to the fraction, then you’ve got a smiley face to the equals.

Layla: So there’s like acircuit there . . .

Carol: . . . between them three so if you change it fraction there it will

go wrong.

The pupils only constructedexpression=numeral statement forms (Figure 5.2), although this is not evidence that they possessed only a place-indicator view of the equals sign. Indeed, given the task goal was to create a classroom resource, they were faithful to how curriculum designers present the equals sign. However, the study evidences a sequential reading of statements from left to right, and an assumption that the on-screen objects should be connected accordingly. The technology enabled them to discover this is not so, and emphasised the need for a “data-flow” conception. The technology also enabled pupils’ “relational thinking” to be explored in a context that did not involve contrived appeals to structure – the “data-flow” conception was independent of statements’ particu- lar numbers and operations. The insight that technology can challenge, and so expose, pupils’ conceptions of notation in novel ways led to a small follow-up study, which is reported in the following section.