The purpose of the pilot trial was to test the robustness and usability of the keypad tools (Figure 10.1) and, assuming it worked as required, to gain initial insights into how children might coordinate balance and substitutive views of equality statements when making diagrams. The trial in fact produced inter- esting data which I report in this section.
The participants were Yaseen and Laura from Study 2 (Section 9.3). The pilot trial was somewhat opportunistic and took place about one month after the trials reported in Study 2. As Yaseen and Laura were already familiar with
Figure 10.1: Keypads for making diagrams
solving diagrams, and had engaged enthusiastically and successfully with the task, I presented them directly with the keypad tools and challenged them to make their own diagrams. Initially I provided keypads with only + operators and later introduced×and then−operators. The total trial lasted 43 minutes. TheSum Puzzlessoftware presents two keypads for making tools (Figure 10.1). The top keypad is for entering a term into the box at the top of each diagram, and in addition to numerals and operators also contains buttons for clearing the screen (“c”) and deleting the left-most and right-most characters in a term (J and I respectively). The bottom keypad is for entering statements and functions similarly to the Equivalence Calculator described in Section 5.2.2. The numeral and operator buttons can be used to enter a term on either side. If the terms are numerically balanced the equals sign appears, as in 2+3 = 3+2; if imbalanced the “not equals” sign appears, as in 2 + 36= 3 + 1; if either side contains an incomplete term nothing appears in the middle, as in 2 + 3 3+. After being shown how to use the keypad gadgets, Yaseen and Laura made eight solvable diagrams during the trial (Figure 10.2). The first diagram (a) contains two statements both of which are substitutive with regard to the boxed term, but only one of which is required to solve it (i.e. produce a single numeral in the box). The second diagram (b) contains four statements, one of which is redundant (30 + 2 = 2 + 30). The next three diagrams (c-e) contain between two and five statements, all of which are required to solve the diagrams. The final three diagrams (f-h) all have more than one type of operator in the boxed term and all contain one or three redundant statements that play no part in
(a) Diagram 1 (b) Diagram 2
(c) Diagram 3 (d) Diagram 4
(e) Diagram 5 (f) Diagram 6
(g) Diagram 7 (h) Diagram 8
solving them. (The redundant statements are 5×5 = 25, 10×5 = 50 and 50 = 10×5 in diagram f; 50 + 8 = 58 in diagram g; and 8×6 = 48 in diagram h).
The children therefore successfully made complex diagrams comprising multiple statements that were both balanced and substitutive with regard to the boxed term. The presence of redundant statements in all the diagrams where the boxed term includes more than one type of operator (f-h) also suggests the chil- dren experienced some challenges ensuring the conservation of quantity across transformations where the ordering of operations was a factor.
The remainder of this section presents three transcript excerpts to illustrate how the children’s discussion provides evidence that they considered and coordinated numerical balance and substitutive meanings for equality statements.
The first excerpt is from the start of the second diagram (Figure 10.2b) and illustrates the children attempting to make a statement that is both balanced and substitutive. Yaseen had put 32 + 54 in the box and wanted to partition the numerals (line 242). Laura suggested 30 + 2 for one side of the statement (line 243) and Yaseen inputted 30 + 2 = 2 + 30 without giving a clear rea- son (lines 246-248). He had entered a numerically balanced statement but, as Laura pointed out (lines 249-251), it was not substitutive with regard to the boxed term. Following this Yaseen attempted to make the statement 32 = 23 (lines 253-257), which is substitutive but imbalanced. Laura then suggested a statement that is both balanced and substitutive (i.e. 30 + 2 = 32, lines 259- 261). Note that Laura intended this statement to partition the numeral 32 suggesting a right to left substitutive reading.
Yaseen: I want to split them two [the numerals in the boxed term] apart. 242
Laura: Let’s do 30 add 2. 243
Yaseen: You can’t do that bit because that’s 2, that’s 30 plus 2. 244
Laura: Oh yeah. 245
Yaseen: [unclear] We can do two more sums, innit? [pause] 2 plus 30, so 246
we want, and then we just place that there. [enters30 + 2 = 2 + 30] 247
Then. . . 248
Laura: I don’t think that one works. 249
Laura: Because it doesn’t give 30 add 2 in the box. 251
R: Do you see what she means, Yaseen? 252
Yaseen: Mm-hm. Wait, 32 then put [unclear] over here, what’s it called? 253
23, and then . . . [inputs 32and 23at the keypad, giving 32 6= 23on 254
screen]. I don’t care if the equals sign’s not there but I know how 255
to do it. Does the equals sign have to be there? [attempts to select 256 326= 23to make a substitution but finds he can’t] 257
R: Laura, have you any ideas how to help? 258
Laura: Um, maybe you can make that sum and then put the answer 259
on the other side, and then, make something that equals 32 and 260
make the other one. 261
R: Okay. Yeah, I think I understand but show us what you mean. 262
Yaseen: 30 plus 2 you’re saying, and what else? Okay, you get that 263
but . . . [inputs30 + 2 = 32 and transforms the boxed term 32 + 54 → 264 30 + 2 + 54] Yeah because we can split that [32] into that [30 + 2]. 265
The second excerpt is from the start of the sixth diagram (Figure 10.2f) and illustrates the children’s awareness of the need for conservation of quantity across transformations. They had entered 15×5 + 9 in the box and Yaseen suggested entering 5+9 = 14 to make a substitution but then decided against it (lines 266- 270). When prompted both children indicated that 15×5 + 9 and 15×14 would give different results (lines 271-274).
Yaseen: 5 plus 9, no you can’t do that. 5 plus 9, because that would make 266
14. 15 times 14. No. [laughs] 267
R: Sorry, what do you mean “no”? 268
Yaseen: Swap it round [i.e. replace5 + 9with14in the boxed term] makes 15 269
plus 14. [he presumably meant 15times14] So that won’t work. 270
R: Why not? 271
Yaseen: It’s going to be different. 272
Laura: [talking at same time as Yaseen] Because then you get another answer. 273 274
The third excerpt is from the start of the seventh diagram (Figure 10.2g) and illustrates the children’s attempts to substitute the term 2−15. Yaseen had entered 58×2−15 in the box and Laura suggested the statement 15−2 = 13 (lines 275-279). Yaseen pointed out that it could not make a substitution in the boxed term because the 15 and 2 are commuted and attempted to input 2−15 = 15−2. He seemed doubtful of its validity and was unsurprised when it appeared on screen as 2−156= 15−2 (lines 280-284). He then suggested the×
operator in the boxed term was the problem (probably alluding to conservation of quantity across transformations – lines 287-288) but Laura pointed out that the statement was imbalanced (lines 286-290).
Laura: I think 15 takeaway 2. 275
Yaseen: 15 takeaway 2? 276
Laura: Yeah, and then . . . 277
Yaseen: Are you sure? 278
Laura: . . . do 13. 279
Yaseen: No, first we got to switch that [i.e. 15−2in Laura’s suggested state- 280
ment] around. [pause] Let’s do 2 takeaway 15. [inputs2−15 = 15−2] 281
Let’s see if that equals, er, recognises it, first. [the statement ap- 282
pears on screen as2−156= 15−2] No, the equals sign isn’t there so 283
we can’t do it. 284
R: Why isn’t the equals sign there? 285
Laura: Because 2 takeaway 15 is minus 13. 286
Yaseen: [talking over Laura . . .] Because the top [boxed term] has a times. So 287
you can’t do, because that one is . . . 288
R: Laura, say that again I don’t think Yaseen heard it. 289
Laura: Um, because 2 takeaway 15 equals minus 13. 290
Yaseen: Yeah, that’s true. 291
In sum, the pilot trial demonstrated that the pupils made predictions about the effect of potential substitutions in an exploratory and reflective manner that took account of the substitutivity of statements with regard to the boxed term (lines 249-251 and 280-284), the numerical balance of statements (lines 286- 290), and the conservation of quantity across transformations (lines 266-274
and 287-288). This enabled the children to confront and discuss issues such as the ordering of operations in 15×5+9 (lines 266-274 and the non-commutativity of 2−15 (lines 275-290) in a way that was purposeful and meaningful to them. In this manner the children engaged with a duality of “is the same as” and “can be substituted for” meanings of the equals sign. This stands in contrast to the previous studies (Chapters 8 and 9) in which pupils, including Yaseen and Laura, engaged exclusively with the substitutive meaning.