Journal for Research in Mathematics Education
2012, Vol. 43, No. 1, 2–33
A Substituting Meaning
for the Equals Sign
in Arithmetic Notating Tasks
Ian Jones
Loughborough University
Dave Pratt
Institute of Education, University of London
We present 3 studies trialing arithmetic tasks that support both substitutive and basic
relational meanings for the equals sign. The duality of meanings enabled children to
engage meaningfully and purposefully with the structural properties of arithmetic
statements in novel ways. In particular, they observed and exploited distinctive transformational effects when using different statement types to make substitutions of
notation. We show how the initial version of the task emphasized the substitutive
meaning but not the basic relational meaning, and an adapted version required both
meanings of the equals sign to be flexibly coordinated. We found that some, but not
all, children were successful at the adapted task and were able to connect making
substitutions of notation with their existing mental calculation strategies.
Key words: Arithmetic; Children’s strategies; Computers; Middle grades, 5–8;
Pre-algebra
The modern equals sign was invented by Robert Recorde in 1557 in The
Whetstone of Witte. He set out his reasoning as follows:
to avoide the tediouse repetition of these woordes : is equalle to : I will sette as I doe
often in woorke use, a paire of paralleles, or Gemowe [twin] lines of one lengthe, thus:
======, bicause noe. 2. thynges, can be moare equalle. (Recorde, 1557, in Cajori,
1928, p. 176)
The equals sign can indicate a variety of things, such as a computational result, as
in 2 + 2 = 4; an identity, as in (x + y)(x − y) = x2 − y2; a function, as in f (x) = cos x;
a substitution, as in x = 1/2; and so on. Digital technology supports further uses of
the equals sign. On traditional calculators “=” appears on a button pressed to get
the result to a programmed sequence of numerals and operator signs. Within
computer programming languages, “=” can assign values or compare two inputs
and return a Boolean result.
The research reported in this article was conducted by the first author under the
supervision of the second author at the University of Warwick, UK. It was funded
by the Institute of Education at the University of Warwick.
Ian Jones and Dave Pratt
3
The focus here is on young children’s conceptions of the symbol “=” in arithmetic
contexts with integers greater than 0. Conceptions of the equals sign are known to
be important for children’s understanding of arithmetic and algebraic notations.
However, most research has been limited to a dichotomy between a naive conception of the equals sign as meaning “write the answer here,” and a more mature
conception of the equals sign as meaning “is the same as.” In this article we argue
for and explore the implications of a broader understanding of the equals sign that
also acknowledges a substitutive meaning.
CHILDREN’S CONCEPTIONS OF THE EQUALS SIGN
Young children from Western countries tend to view the equals sign as a place
indicator meaning write the answer here (Behr, Erlwanger, & Nichols, 1976;
Kieran, 1981; Knuth, Stephens, McNeil, & Alibali, 2006; Renwick, 1932; Warren,
2003). This is evidenced by children accepting only statements of the type expression = numeral, and rejecting other types such as expression = expression (Behr
et al., 1976). This place-indicator view of the equals sign works fine in some arithmetic situations, such as 3 + 4 = 7, but can lead to errors such as “running” statements, as in 3 + 4 = 7 + 2 = 9 (Hewitt, 2001; Renwick, 1932; Sáenz-Ludlow &
Walgamuth, 1998). More generally, an exclusive place-indicator view leads to
inflexible thinking about arithmetic (Baroody & Ginsburg, 1983; Li, Ding, Capraro,
& Capraro, 2008; McNeil & Alibali, 2005), and later difficulties with algebraic
equation solving (Knuth et al., 2006). The place-indicator view is also resistant to
change (McNeil & Alibali, 2005).
Nevertheless, there is strong evidence that such problems can be avoided or, with
effort, overcome by teaching children that “=” means “is the same as” and allowing
them access to a wide variety of statement types (Baroody & Ginsburg, 1983;
Carpenter, Franke, & Levi, 2003; Li et al., 2008; Linchevski & Livneh, 1999;
Molina, Castro, & Mason, 2008; Pirie & Martin, 1997; Rittle-Johnson, Matthews,
Taylor, & McEldoon, 2011). This leads to more flexible thinking about arithmetic,
including statement types not encountered previously (Baroody & Ginsburg, 1983;
Li et al., 2008). Children who consider the equals sign to have a numerical sameness
meaning and are accepting of a wide variety of arithmetic statement types are said
to possess a basic relational view of the equals sign (Baroody & Ginsburg, 1983).
More recent interventions have sought to promote more sophisticated conceptions
of arithmetic equivalence, referred to as a full relational view (Carpenter et al.,
2003; Molina et al., 2008). A key idea is to encourage children to exploit the structural properties of carefully selected and sequenced statements such that truthfulness can be established without knowing what the number is on each side. This is
typically achieved by appealing to arithmetic principles within the structure of a
given statement. Some examples, adapted from Molina et al. (2008), are shown in
Table 1.
The development of a full relational view is necessary for moving from understanding arithmetic equivalence to understanding algebraic equivalence (Kieran,
4
A Substituting Meaning for the Equals Sign
Table 1
Examples of Statement Types That Might Be Presented to Learners to Promote a Full
Relational View
10 + 4
=
4 + 10
Commutative property of addition
15 – 5
≠
5 – 15
Noncommutativity of subtraction
100 – 94 + 94
=
100
13 + 11
=
12 + 12
Inverse relation of addition and subtraction
Compensation relation
1981; Linchevski & Herscovics, 1996; Knuth et al., 2006). Within algebra, the
equals sign carries a multiplicity of meanings, such as those given in the introduction to this article. A particularly important meaning is that of substitution (Collis,
1975; Kieran, 1981; Linchevski & Livneh, 1999). The development of a full relational view, including an understanding of the substitutive aspect of equivalence
relations, is closely connected to the particular symbols used and the actions
intended. For example, statements such as i2 = –1 can be interpreted as a rule for
substitution, namely that i2 can be substituted by –1 (and vice versa) in all equations. In terms of learner development, Filloy, Rojano, and Solares (2010) showed
that students who can readily substitute a number for an unknown often struggle to
substitute an expression containing a second unknown.
Our interest here is how the substitutive meaning for the equals sign might be
implemented and supported in arithmetic notating tasks, by which we mean
activities designed to promote learning about, and learning through, properly
formed arithmetic notation. In the remainder of the article we distinguish between
three views of the equals sign: place-indicator, substitutive, and basic relational.
We consider that substitutive and basic relational are both important aspects of full
relational view. However, within a purely arithmetic context, lacking unknowns, it
is not immediately obvious how substitution might be relevant to designing educative notating tasks. In the following section we provide a pedagogic argument for
an arithmetic substitutive meaning and describe one way that it can be implemented.
DIAGRAMMATIC REASONING
Our pedagogic argument for supporting the substitutive meaning in arithmetic
notating tasks draws on Dörfler’s (2006) interpretation of 19th-century philosopher
Charles Sanders Peirce’s (1991) diagrammatic reasoning. Dörfler highlighted the
educational implications of Peirce’s attempt to resolve the paradox between deductible and discoverable aspects of mathematics by emphasizing the importance of
formal inscriptions in the development of mathematical and scientific thinking.
Damerow (2007) traced those aspects of diagrammatic reasoning that can be seen
in the development of arithmetic. For example, the introduction of Arabic numerals
aided calculation and made new kinds of mathematics possible. From the Peircean
viewpoint, a “diagram” can be a geometric construction, a graph, or a system of
Ian Jones and Dave Pratt
5
simultaneous equations, and so on. In one sense, this is a looser use of diagram
than everyday associations with figures and pictures suggests, but in another sense
it is more restrictive, referring only to those inscriptions that form precise mathematical structures with agreed-upon rules for making transformations.
Dörfler’s attention to diagrammatic reasoning began with his argument that there
exists within mathematics education a widespread, but rarely stated, assumption
that mathematics is not directly accessible. Learners must instead deal with representations, including the conventional symbols of arithmetic, but the mathematical
objects themselves are concealed behind these visible inscriptions. This, argues
Dörfler, is why so many people feel alienated when they try to learn mathematics:
They often experience that they do not get close to those genuine object which mathematics purportedly is all about, they belief [sic] they lack the necessary abilities to
think “abstractly”, they are convinced that they do not understand what they are
expected to understand. They want to reach through the representations to the abstract
objects but without success. (p. 100)
Dörfler (2006) put forward an alternative, diagrammatic view in which doing
symbolic mathematics is based in making rule-governed transformations of visible
representations. Mathematics learning can then be seen as an empirical and experimental activity involving the observation and comparison of inscriptions. The focus
is not on accessing hidden Platonic entities but on potential actions with what can
be seen on the page or computer screen. A similar, non-Platonic focus on mathematical symbols and their transformations being at the very heart of mathematics is
apparent in the work of Wittgenstein (1967). The educative value of the diagrammatic approach to task analysis and design is that it allows learners to construct
meaning from seeing actual and potential transformations of visible inscriptions
toward a task goal. This stands in contrast to meaning arising from specified referents
for symbols (Kirshner, 2001), and to the “meaningless” proceduralism commonly
associated with learning formal symbol systems in mathematics classrooms. In the
following section we set out one way that the substitutive meaning for the equals
sign can support diagrammatic reasoning about equality statements.
SUBSTITUTIVE AND BASIC RELATIONAL MEANINGS
FOR THE EQUALS SIGN
In this section we describe specially designed, software-based arithmetic tasks,
called Sum Puzzles1, that support both the substitutive and basic relational meanings for the equals sign. We then describe how they can enable an exploratory
diagrammatic approach to learning formal symbolism.
The software presents learners with a sequence of arithmetic puzzles, an example
of which is shown in Figure 1. The version used in the studies contained no instructions or prompts; these were provided by the researcher. Sum Puzzles is operated by
1 To aid clarity, the screenshots of Sum Puzzles in this article are recreations using the latest version
of the software.
6
A Substituting Meaning for the Equals Sign
Figure 1. A computer-based task that supports a substituting meaning for the equals sign.
interacting directly with the notation on the screen, and there are two key functions
for solving puzzles. The first is selecting a statement, which is done by clicking on
its equals sign. The selected statement then stays highlighted until it has been used
to make a substitution, or another statement is selected instead. The second key
function is attempting to make a substitution, which is done by clicking on one of
the symbols within the boxed expression at the top of the screen. If a statement is
selected, and is substitutive with regard to (i.e., can be substituted for) the clicked
symbol within the expression, then a substitution takes place in the expression. If a
statement is not selected, or a statement is selected but is not substitutive with regard
to (i.e., cannot be substituted for) the clicked symbol, nothing happens.
Consider the puzzle in Figure 1. The particular numerals and their positions
within the puzzle are not of importance here; the purpose is to illustrate how potential substitutions are identified and then carried out. The task goal is to use the
provided statements to transform the boxed expression (31 + 40) into a single
numeral (i.e., its “answer,” 71). For example, we might start by selecting the statement 31 = 30 + 1 and use it to make a substitution of the numeral 31 in the boxed
expression, and so transforming 31 + 40 → 30 + 1 + 40. We can carry on using the
statements to make substitutions in this manner until the goal state (71 in the box)
is achieved. An example solution for achieving this for the puzzle shown in Figure
1 is provided in Table 2.
The task affords diagrammatic ways of viewing, working with, and talking about
arithmetic notation compared to typical approaches in the literature. The key
Table 2
An Example Solution Presented Statement by Statement for the Puzzle Shown in Figure 1
Statement used
Resultant boxed expression
31 + 40
31
=
30 + 1
30 + 1 + 40
30 + 1
=
1 + 30
1 + 30 + 40
30 + 40
=
70
1 + 70
1 + 70
=
71
71
Ian Jones and Dave Pratt
7
affordance is that of visually scanning a puzzle for matches of notation to ascertain
where substitutions can be made. For example, we might start by noticing that the
boxed expression in the puzzle in Figure 1 contains the numerals 31 and 40, and so
look for occurrences of these numerals within the statements. This is a qualitatively
different way of looking at arithmetic notation from performing calculations. It is
diagrammatic in the sense described in the previous section because it focuses
attention on the written symbols themselves rather than the arithmetic objects and
principles they might reference.
Another affordance is describing and predicting the distinctive substituting
effects of different statement types. For example, in Figure 1, if the statement 31 =
30 + 1 is used to make a substitution, then the boxed expression on the screen can
be observed transforming from 31 + 40 to 30 + 1 + 40. In this sense, the statement
31 = 30 + 1 can be seen to partition the numeral2 31. (We use partition here to mean
the transformation of one integer into the sum of two integers.) More generally, a
statement’s transformational effect often depends both on its structure and the notation it transforms. For example, 30 + 40 = 70 has a compositional form if used to
substitute for 30 + 40, and a partitioning form if used to substitute for 70. Similarly,
the statement 30 + 1 = 1 + 30 in Figure 1 can be seen to change the order of the
numerals 30 and 1 when used to make a substitution in the boxed expression. In
this way, observing and predicting distinctive substituting effects can offer learners
engagement with the structural properties (partitioning and commuting in the above
examples) of different statement types in terms of observable and potential actions
on symbols.
The careful selection of statements when populating puzzles can allow the
designer to draw attention to particular principles and conventions. In Figure 1,
for example, a left-to-right reading of statements is assumed in which the object
on the left of the equals sign is always replaced by the object on the right of the
equals sign. (This left-to-right reading was intended to help students get started
with solving puzzles. As will be seen subsequently, some students were oblivious
to directionality when working with the software.) The partitioning statement
31 = 30 + 1 affords place-value decompositions into two numerals, and the
commutative statement 30 + 1 = 1 + 30 affords swapping around two adjacent
numerals.3
The software can also be set up to allow learners to try making their own puzzles
using a specially designed “keypad tool” (Figure 2). It was designed to resemble
a simple calculator and is used for inputting equality statements in order to make
2 This use of numeral rather than number is consistent with Dörfler’s (2006) emphasis on concrete
symbols rather than abstract concepts. When using the software, participants observed “31” split into
“30” and “1.”
3 Note that puzzles and associated strategies for solving them need not be like that exemplified in
Figure 1. For example, we might include distracter statements that are not needed in order to solve the
puzzle. Also note that the task goal need not necessarily be to replace the given expression with its
“answer.” Figure 1 might as well have begun with 71 in the box and the alternative instruction “Make
substitutions until you have changed the number below into 31 + 40.” The options when designing
puzzles are very open and, in principle, limitless.
8
A Substituting Meaning for the Equals Sign
Figure 2. Keypad tool for entering statements when making a puzzle.
puzzles. It lacks an equals button, and instead, each side of a statement is entered
at a separate keypad. If the two sides have the same value, then “=” appears
between them; otherwise “≠” appears, as shown in Figure 2. Once a true statement
has been constructed, it can be placed anywhere in the puzzle by clicking at the
desired position. False statements cannot be placed in the puzzle, and if this is
attempted, nothing happens. A similar tool, but with one keypad rather than two,
exists for entering expressions. (The keypad tools proved to be somewhat slow
and have since been removed. Expressions and statements are now typed in at the
computer keyboard.)
In terms of equivalence, there is an important difference between the activities
of solving puzzles and making puzzles. Puzzle solving emphasizes the substitutive
meaning instead of the basic relational meaning of the equals sign. This is because
the statements are presented as rules for making substitutions and their truthfulness is irrelevant. The false statement 31 = 30 + 2 can be used to exchange symbols
just as well as the true statement 31 = 30 + 1. The difference is that the former
would lead to mathematical nonsense. Nevertheless, truthfulness, and therefore
the basic relational meaning of the equals sign, need not concern us when solving
a puzzle. In contrast, puzzle making emphasizes both the substitutive and basic
relational meanings. This is because the keypad tool allows only true statements
to be entered, and statements must be constructed such that both sides have the
same value. Moreover, for the puzzle to work, the statements entered must be
substitutive with regard to the boxed expression at the top of the puzzle. In this
way, puzzle making requires thinking of the equals sign both in terms of sameness
and substitution when constructing statements.
RESEARCH FOCUS
The purpose of the research reported in this article was to test the design affordances described in the previous section. In particular, we wanted to find out
whether children can work with and talk about arithmetic notation in diagrammatic
ways that are qualitatively different from how they work and talk when computing
arithmetic problems. It was not an explicit aim to test whether sustainable learning
gains would be achieved, or whether learning was transferable to different contexts
or nondigital media.
9
Ian Jones and Dave Pratt
RESEARCH QUESTIONS AND EVIDENCE SOUGHT
Three studies were conducted with children aged 9 to 12 years to address the
research questions shown in Table 3. Each study consisted of two or three
substudies, which we refer to here as trials. Each trial was a single observed use of
the software and consisted of one pair of children working with the software for a
period of time. Study 1 addressed the first question and comprised two trials, Study
2 addressed the second question and comprised a further two trials, and Study 3
addressed the final two questions and comprised three trials (14 children in seven
pairs altogether). The first question was fundamental to all three studies because it
sought to test the underlying design affordances described in the previous section.
As such, it was informed by elements of the data from all seven trials as shown in
Table 3. The second research question tested a specific observation that the substitutive meaning of the equals sign is emphasized and the basic relational meaning
is de-emphasized when children solve puzzles. The final two research questions
built on this finding, and we adapted the design to help children emphasize each
meaning of the equals sign flexibly and appropriately towards a specified task goal.
This involved challenging the children to make their own puzzles, which requires
using both meanings of the equals sign. In this sense, we aim to present the four
research questions as a coherent and cumulative investigation into the potential of
the substitutive meaning for the equals sign for designing arithmetic notating tasks.
Data from Trials 1 to 4 have been published previously as isolated studies designed
to address questions 1 and 2 (Jones, 2007, 2008). Here we draw mostly upon data
from Trials 5 to 7, results of which have not been published previously. When it
Table 3
The Three Studies and the Research Questions They Addressed, Along With Details of the
Trials That Informed the Studies
Study
Research question
Trials
Participants
Task
1
How can the substitutive
meaning for the equals sign
promote attention to statement
structure?
1–7
9 and 10 years
High and midachievers
Solving
puzzles
2
Are the substitutive and basic
relational meanings for the
equals sign pedagogically
distinct?
3, 4
9, 10, and 12
years
High achievers
Solving
“nonsense”
puzzles
3
How can children appropriately
emphasize substitutive and
basic relational meanings for
the equals sign?
5–7
9 and 10 years
High and midachievers
Making
puzzles
How can children connect their
implicit arithmetical knowledge
with making substitutions?
5, 6
9 and 10 years
High and midachievers
Making
puzzles
based on
strategies
10
A Substituting Meaning for the Equals Sign
aids clarity, or when no relevant data are available from these last three trials, we
also draw on data from the first four trials to inform questions 1 and 2.
We sought qualitative evidence for each research question. For question 1 we
analyzed what children said when solving puzzles. In particular, we looked for
articulations of partitioning effects using language such as splits up and separates,
and articulations of commutative effects using language such as swaps and switches
around. We also looked for the articulation of visual searches for potential substitutions, such as (for the puzzle in Figure 1) look for another 31. Given that question
1 addresses the underlying rationale of our approach to task design, we expected
such language to permeate all the studies, even when the explicit focus was on other
research questions.
For question 2 we presented children with sequences of puzzles that contained
some false equality statements. (Note that the keypad tool shown in Figure 2 does
not allow this, so the researchers programmed the false equalities directly into the
puzzles.) We anticipated that the children would not comment on the fact that the
puzzles they were solving contained false equalities, or that the substitutions made
using them would lead to mathematical nonsense. This would suggest they were
not concerning themselves with the basic relational view of the equals sign, in
which both sides have the same number. We also expected that, when asked posttrial, the children would not be able to comment on whether false equalities had
been present in any of the puzzles.
For the final two questions we were interested in seeing whether children were
able to make their own puzzles (question 3) and whether they could make puzzles
that reflected their own addition strategies (question 4). Making puzzles requires
emphasizing both the substitutive and basic relational meanings of the equals sign,
as appropriate. We also analyzed the children’s discussion to see when they were
articulating the substitutive view, as described previously for question 1, and when
they were concerned with the numerical balance across statements, which would
be characterized by discussion of calculations.
METHOD
The research questions set out in the previous section were informed by trials
of computer-based tasks with pairs of children, using a method adapted from
Noss and Hoyles (1996). Noss and Hoyles described how they created windows
on mathematical meanings through the provision of digital tools in a microworld
environment. More specifically, Logo-based microworlds in their studies enabled
research subjects to gain insight into powerful mathematical ideas and, at the
same time, afforded the researchers the opportunity to draw inferences from the
subjects’ discussion and use of digital tools. The second author has previously
adopted this approach through the design of a learning environment that promotes
stochastic abstraction (Pratt, 2000). The present study represents a further adaptation of the approach in that the Sum Puzzles software is more constrained than
in those previous studies, focusing tightly on diagrammatic reasoning associated
Ian Jones and Dave Pratt
11
with the equals sign. Nevertheless, the approach does exploit the capacity for
Sum Puzzles to provide both the children and the researchers a window on this
aspect of mathematics.
This approach enabled the production and capture of rich, qualitative data akin
to semistructured interviews of pairs of students that aim to stimulate their discussion (see Evens & Houssart, 2007). Such data are richest when children listen,
explain, and come to agreement when working on tasks. However, the quality of
talk is highly dependent on classroom culture and previous educational experiences
(Mercer & Littleton, 2007), and the development of such a culture is beyond the
scope of short trials. As such, the researcher played a participatory role during trials
to prompt explanations (e.g., “Why do you think that?” “Can you explain what you
did there?”) and to offer encouragement (e.g., “You’re almost there”), as well to
deal with technical problems. The software was introduced to each pair of children
by telling them the task goal (transform the boxed expression into its answer) and
familiarizing them with the two functionalities of selecting a statement by clicking
on its equals sign and trying to make a change in the boxed expression by clicking
on it.
Seven trials were conducted, each lasting 40–90 minutes. Six of the trials were
conducted with Year 5 children (ages 9 and 10) from three different schools. There
were three reasons for choosing this age group: (a) Algebra is not normally introduced in the United Kingdom until Year 7, and the school years prior to this are of
key interest for presenting arithmetic in more “algebraic” ways; (b) typical Year 5
children can be expected to use implicit arithmetic principles in their mental addition strategies, which was required for the task used to address question 4 (see
subsequent description); (c) there are no mandatory National Curriculum assessments in England during Year 5, making access during school hours easier than for
other years. On the basis of previous pilots of the software with adults and children
known to the authors, we were confident that children this age would not find
solving and making puzzles easy. Five of the trials were repeated at least once with
high- and medium-achieving children, as judged by their class teachers, in order to
explore anticipated individual variations in performance. Two of the trials (Study
2) involved only high-achieving children because we required participants with
confident computational skills. Low-achieving children were not requested,
because they could be expected to struggle with aspects of the task such as reading
notation and performing two-digit mental arithmetic (Department of Children,
Schools and Families, 2007). One of the trials used to inform question 2 (see
previous section) was conducted with high-achieving Year 8 children (ages 12 and
13) to maximize the possibility of the participants noticing the presence of false
equalities in the puzzles presented during the trial.
In each trial, a pair of children worked together at a single computer. Their
onscreen interactions and discussion were recorded as audiovisual movies, and the
resulting data were processed and analyzed in three stages. The first stage was
transcription using the qualitative analysis software Transana, which dynamically
links text to movie data and so enables the analyst to stay close to and move between
12
A Substituting Meaning for the Equals Sign
both the textual and multimedia representations of the data. As such, the data are
more accessible than transcripts alone, which are less rich than multimedia data,
or movies alone, which must be viewed in real time. Second, a trace of each trial,
which is a text-based chronological description of key events that avoids judgments
and interpretation as far as possible, was produced (Pratt, 1998). A trace is an
evolutionary step from a transcript toward a written report and comprises plain
prose interspersed with transcript excerpts and, where helpful, screenshots of the
children’s activity. Constructing a trace requires the analyst to move between overviews and fine-grained views of the data in order to identify cumulative, key events
over the duration of a trial. Third, each line of the transcripts of children’s talk was
coded for children discussing (a) computational results of expressions and statements (e.g., 12 add 1 makes 13 ), (b) visual matches of notation when predicting or
describing where substitutions could be made (e.g., look for another 13 ), and (c)
the distinctive transformational effects of different statement types when imagining
or making substitutions (e.g., that split it up). Most instances of the children’s talk
that were coded fell unambiguously into these three categories and closely resembled the examples given. Transcript lines that did not fall unambiguously into the
preceding categories were not coded. All coding was carried out by the first author,
and examples and interpretations were checked in meetings with the second author.
The three codes emerged from the data over the first two trials and are tied closely
to the predictions of question 1 (see previous section), but proved valuable for
analyzing later trials, too. For example, question 2 sought to replicate the data for
question 1 using a slightly modified task. Note that children’s discussion, rather
than onscreen interactions, was coded, although the latter were essential to interpreting the discussion, and Transana’s dynamic linking between text and movies
greatly assisted in this. Transana also enables the production of time-sequenced
graphical displays of coding (see Figure 5), which provides the analyst with a visual
overview of the amount, density, and sequencing of coding over the duration of a
trial. This allows pattern matching by eye, such as looking for groupings of codes
within a trial and which codes tend to precede others.
THE STUDIES
The research was conducted as a series of three studies comprising two or three
trials each. In this section we present data to inform each of the studies in turn.
Study 1: How can the substitutive meaning for the equals sign promote attention
to statement structure?
In Study 1 we sought to determine whether children articulated visual searches
for matches of numerals and the distinctive substitutive effects of different statement types. This was a test of whether the basic diagrammatic design principles
would support engagement with notational structure and was informed by evidence
from all seven trials. Here we present representative transcript excerpts from across
the trials to exemplify common features of the children’s talk, and a graphical
Ian Jones and Dave Pratt
13
overview of the children’s puzzle-solving activities across all seven trials to illustrate the relative proportions of these features.
The first feature common across the seven trials was children attempting to
determine where allowable substitutions could be made by visually searching for
matching pairs of symbols on the computer screen. The following representative
excerpt is from Trial 5 and occurred about 10 minutes into the trial.4 It illustrates
2 girls looking for pairs of the numerals 19 and 31 when trying to solve the puzzle
shown in Figure 3.
Figure 3. A puzzle presented to children in Trials 5, 6, and 7. It is typical of the puzzles
presented in all three studies and contains three different types of statements (c = a + b,
a + b = b + a, a + b = c), which are loosely grouped together.
1. Bridie: 31 plus 19.
2. Nadine: 19. What’s that?
3. Bridie: 31 . . . look for a 31 somewhere.
4. Nadine: Well, I found a 19 and another 19.
5. Bridie: But we need something that will equal 19. Aha, I found a 31.
The second feature was children describing and predicting the distinctive substituting effects of statements of the form a + b = b + a or c = a + b in terms
of commuting and partitioning, respectively. We first present an example
of commuting. The researcher (R ) asked the children in Trial 6 to explain why
27 + 23 = 50 had failed to transform 23 + 27. They said that the numerals were “the
other way round” (line 12) and then used 23 + 27 = 27 + 23 to commute the
numerals.
6. R:
What have we got in the green box?
7. John: 23 add 27.
8. R:Can you see that anywhere in the puzzle? Anywhere outside the green box.
9. John: 23 add 27, equals 50. That’s it.
10. R:
Anywhere else?
4All
participants’ names in this article are pseudonyms.
14
A Substituting Meaning for the Equals Sign
11. John: Um, there?
12. Derek: Yeah, but 27 add 23 is the same. But that’s just the other way round.
13. John: Yeah, I know. So, and this one as well.
14. Derek: 23 add 27 and that’s the same as that.
15. John: Yeah, so try this one.
16. Derek: 27 add 23.
17. John: Try this one, try this one. Try this one, it’s got the same. There you go. Just
changed it round.
Next we present an example of partitioning taken from Trial 1, in which a pair
of children had been presented with the puzzle shown in Figure 4. One of the children quickly identified how to begin solving a puzzle by predicting the partitioning
effects of a statement (41 = 40 + 1):
18. Terry: Oh! That’s the one that you do first! It has to be.
19. R:Why?
20. Terry: Because it’s splitting up the 40 and the 1.
There were also occasions in every trial when the children did not identify where
substitutions could be made or exploit the distinctive substituting effects of different
statement types, but instead focused on computing results. The following example
occurred 10 minutes into Trial 6:
21. John: 9 add 12 add 1 equals 22.
22. Derek:21.
23. John: No, it’s 22. 13 add 9.
24. Derek: Hm, no, 9 add 12. 9, 13 add 12. No, 13 . . .
25. John: 12 add 1 is . . .
26. Derek: Yeah, 22 because it’s 9 add 12 add 1 is 22.
Figure 4. A puzzle presented to children in Trials 1 and 2.
In every case, such computation was a distraction from reaching the task goal,
which depends solely on identifying and making substitutions. Overall, the children
engaged more with the substitutive properties of arithmetic statements than they
did with computing results. This can be seen in Figure 5, which shows a Transanaproduced, time-sequenced map of the children’s puzzle-solving activities across
the seven trials. Time from 0 to 40 minutes is shown on the horizontal axis, and
Figure 5. Time-sequenced coding of children’s puzzle-solving activities for each trial. Each row shows
a single trial, and the horizontal axis shows time in minutes. Coding of each line of the transcript of
children’s talk is displayed as blocks and reflects children discussing the computation of results, visual
matches of notation, commuting effects of substitutions, and partitioning effects of substitutions. The
diagram enabled us to search visually for relative proportions of each code across all seven trials.
Ian Jones and Dave Pratt
15
16
A Substituting Meaning for the Equals Sign
each row shows a pair of children. Within each row the relative proportions of
codings of the children’s discussion can be seen as blocks, each of which represents
an instance of children computing results (compute), searching for potential substitutions (match), or describing or predicting commutative substitutions (commute)
or partitioning substitutions (partition). For example, in Trial 1, just before the 5th
minute, the children’s talk was coded as compute, followed by match. Note that
Figure 5 gives only an approximate sense of the relative proportions of these four
kinds of incidents and is not intended as a measure of the amount or quality of each.
For example, a block in the Commute row might simply reflect one child describing
a substitution as “swapping round”; another of similar length might reflect two
children discussing how to commute two numerals as part of a sophisticated
strategy for solving a puzzle. Nonetheless, Figure 5 illustrates how relatively infrequently across the trials the children computed results. The notable exceptions to
this are Trial 5, in which two particularly enthusiastic children seemed keen to
impress the researcher with their ability to compute results quickly, and the long
compute block in Trial 4 when the children responded to the researcher’s questions
about the truthfulness of the onscreen statements after they had solved all the
puzzles in the sequence. Compute is still present to a larger degree in Trials 6 and
7 than Trials 1 to 3, but less so than the substitutive codes of match and commute
combined. Talk was coded less for “partition” than for “commute” in every trial,
and the children in Trials 6 and 7 did not articulate partitioning substitutive effects
at all. The role of articulating partition when working with the software is considered later in the paper.
In sum, Study 1 demonstrated that in a task that placed emphasis on the substitutive meaning, the children appeared to pay attention to statement structure by
referring frequently to matching pairs of identical symbols, the “swapping” property of commutative statements and, to a lesser extent, the “splitting” property of
partitioning statements.
Study 2: Are the substitutive and basic relational meanings for the equals sign
pedagogically distinct?
The second study was designed to test an emerging hypothesis that when solving
puzzles, children tend to emphasize a substitutive meaning at the expense of a basic
relational meaning. This is not to say that the children in Study 1 viewed the equals
sign only as substitutive at all times, but that they were commonly indifferent to
the truthfulness of the statements provided in the puzzles. Indeed, the most efficient
way to solve puzzles is to look for substitutions that can be made and not concern
oneself with the truthfulness of the presented statements. To test this we designed
a sequence of puzzles that included false equalities. The findings from Study 2 have
been published previously (Jones, 2008) and are briefly summarized here.
The 11 puzzles designed for Study 2 were similar to those used in Study 1, except
that the last 4 contained false equalities. These began with subtle falsehoods, such
as 15 + 28 = 44 in Puzzle 8, and proceeded to what many would recognize as blatant
absurdities, such as 77 = 11 + 33 in Puzzle 11. The puzzles were trialed with a pair
Ian Jones and Dave Pratt
17
of children aged 9 and 10 (Trial 3), and another pair both aged 12 (Trial 4). In order
to ensure that the arithmetic statements were within the computational ability of
the children, and therefore the false equalities would be readily noticeable to them,
the children selected were all determined by their teachers to be high achievers in
mathematics. During the two trials, the presence of false equalities had no observable effect on how the children talked about and worked with the puzzles. Neither
did any of them comment on their presence until they had solved, or almost solved,
the final puzzle, whereupon the result “23” appeared on screen as the solution to
143 + 77. Only at this point did a child in each trial comment, with some surprise,
that some of the equalities were incorrect. When asked whether any of the previous
puzzles had included false equalities, none of the children was able to say (three of
the previous puzzles had included false equalities).
There were two outcomes of Study 2. First, the substitutive and sameness meanings for the equals sign are distinct, because when puzzle solving, the children made
substitutions using false statements but were not aware of their falsity. Second,
although the children articulated the distinctive substitutive properties of different
statement types, what they were doing on the screen was mathematical nonsense.
Only when it was taken to extremes in the final puzzle (e.g., 77 = 11 + 33) did they
notice this, and even then—in both trials—only when they had been working on
the puzzle for some minutes. In sum, the children were engaged with making
substitutions within the context of puzzle solving but were not engaged with the
numerical sameness of statements or the conservation of quantity across transformations of the boxed expression. The puzzle-solving task emphasized the substitutive meaning at the expense of the basic relational meaning—they were playing the
“substitution game” without consideration of the meaningfulness of their activities.
As such, our next step was to redesign the task so that children would be required
to emphasize both the substitutive and sameness meanings of the equals sign when
working toward a specified goal. This was the focus of Study 3.
Study 3
The substitutive and basic relational views of the equals sign are both important
aspects of a full relational view, yet the puzzle-solving task appeared to emphasize
the former at the expense of the latter. Although the children computed results when
solving puzzles, as shown in Figure 5, this was always an aside from working toward
the task goal because solving puzzles does not require a basic relational view, as
demonstrated by Study 2. Study 3 was designed to find out whether children could
emphasize the two meanings appropriately by ensuring that the task goal required
both. To this end the children were challenged to make their own puzzles, which
required inputting arithmetic statements that were both true (basic relational
meaning) and that could be used to transform the boxed expression (substitutive
meaning). This generated evidence used to inform the final two research questions:
How can children appropriately emphasize substitutive and basic relational meanings for the equals sign? How can children connect their implicit arithmetical
knowledge with making substitutions?
18
A Substituting Meaning for the Equals Sign
Three trials were conducted with pairs of children aged 9 and 10 (Trials 5 to 7).
Each trial entailed two or three sessions and contained three components.
Component 1 involved puzzle solving and was designed to familiarize the children
with the software and the nature of the puzzles. This did not directly inform the
research questions of Study 3, but we use it below to draw up qualitative profiles
of the children involved. Component 2 involved unstructured puzzle making and
was designed to inform one of the research questions. Component 3 involved
structured puzzle making and was designed to inform the other research question.
We address each component in turn.
Component 1: Qualitative profiles of the children. The children in Trial 5 (Bridie
and Nadine) were described by their teacher as high achievers in mathematics. They
were confident with arithmetic and eager to impress the researcher throughout the
trial with their quickness of calculation, as reflected by the high number of compute
codings in Figure 5. They were immediately accepting of varied statement types,
suggesting a basic relational view of the equals sign. For example, the first puzzle
presented to them contained 13 = 12 + 1, and the researcher asked about each side,
to which Nadine replied “12 and a 1. So it makes there be another 13.” When
working through puzzles, they exploited commutative and partitioning effects and
solved them efficiently, again reflected in Figure 5. This suggests that they readily
viewed the equals sign substitutively, as required to achieve the task goal.
The children in Trial 6 (Derek and John) were described as middle achievers in
mathematics by their teacher. The data provided substantially more insights into
John’s thinking because he was more talkative than Derek. John was hesitantly
accepting of statement types other than expression = numeral, saying of 12 + 1 =
1 + 12 in the puzzle presented to them: “It’s quite a different sum, isn’t it? Like 12
add 1 equals 1 add 12.” This suggests that he accepted a basic relational meaning
of the equals sign, although we are unable to comment on whether Derek shared
this view. Their puzzle-solving activity was dominated by computing results and
looking for notational matches to establish where substitutions could be made, as
shown in Figure 5. John often articulated the commutative effects of substitutions,
such as in lines 6–17 (previously presented), but not partitioning effects. Moreover,
these articulations of commutation tended to be descriptive and, unlike for the
children in Trial 5, were not exploited strategically to achieve the task goals. This
suggests that John viewed the equals sign substitutively at times but did not fully
grasp its usefulness for solving puzzles, as reflected by his dismissive comment
“Just changed it round” in line 17. Derek struggled to continue when John left the
trial for 10 minutes (to have his school photograph taken), and we suspect he did
not view the equals sign substitutively.
The children in Trial 7 (Colin and Imogen) were described as middle achievers
in mathematics by their teacher. They were reticent compared with the children in
the other six trials and solved puzzles with some difficulty. It is not possible to
determine whether they viewed the equals sign as a basic relation, and Imogen in
particular tended to express calculation strategies when she did talk. Colin, and to
Ian Jones and Dave Pratt
19
some extent Imogen, also looked for matches of notation in order to attempt substitutions, as can be seen in Figure 5. Commutation was articulated at times, but this
was generally in response to researcher prompting (“What happened when you
clicked in the box?” and so on), and the commutative effects of substitution were
only once explicitly used toward achieving the task goal (Imogen: “If you try and
click on one of the ones that swap them round, it might work so you can click on
that one”). Otherwise their puzzle solving was characterized by trial and error with
little evidence of predicting and exploiting the substitutive effects of different statement types. We therefore suspect that Colin and Imogen only rarely viewed the
equals sign substitutively, if at all.
Having summarized the work of the three pairs of children based on Component
1, we now turn to evidence from Components 2 and 3 in order to inform the third
and fourth research questions.
Component 2: How can children appropriately emphasize substitutive and basic
relational meanings for the equals sign? Puzzle solving requires only emphasizing
the substitutive meaning, as shown in Study 2. Puzzle making, however, requires
emphasizing both meanings: The basic relational meaning is necessary to ensure
that statements entered are true, and the substitutive meaning is necessary to ensure
that statements can be used to transform the boxed expression. For Component 2
of the trials children were provided with keypad tools (Figure 2) and challenged to
make their own puzzles. We were interested in whether they could make solvable
multistep puzzles, by which we mean puzzles that require more than one substitution to be made in order to solve them. For example, a puzzle comprising, say, the
boxed expression 30 + 41 and the single statement 30 + 41 = 71 is solvable but only
requires one substitution to be made and so is single-step.
Bridie and Nadine (Trial 5) were notably more successful at making their own
puzzles than the children in Trials 6 and 7. In total they made four solvable multistep
puzzles. An analysis of their discussion revealed that they initially viewed the equals
sign as a basic relation when deciding which statements to use in their puzzles. For
example, they began making their first puzzle by entering 25 + 1 and then, in
discussion with one another, entered the three statements 2 + 24 = 24 + 2, 11 + 15
= 15 + 11 and 25 + 1 = 1 + 25. These statements conformed to the basic relational
meaning of the equals sign (they were true) and contained terms that equaled the
boxed expression (25 + 1), but only one of them was substitutive with regard to the
boxed expression. The data do not reveal why they chose three commutative statements, although it is likely they remembered the presence of such statements in the
puzzles they had solved previously.
At this point the researcher asked them to test whether the puzzle could be solved.
Nadine used 25 + 1 = 1 + 25 to change the boxed expression into 1 + 25.
27. Nadine: It just switches it round.
28. Bridie: At the minute it’s impossible to do.
29. Nadine: You can’t do it. We need to have a sum that’s only one thing.
20
A Substituting Meaning for the Equals Sign
Here the girls realized the need for statements to be substitutive with respect to
the boxed expression. However, after some brief discussion they entered 23 + 3 =
20 + 6, which was true and equal to the value of the boxed expression but still not
capable of making a substitution. Nadine then realized statements needed to be
matched with the expression in order to be substitutive: “Let’s make one of them
that says 25 add 1 because then it will be easy to do.” She entered 25 + 1 = 19 +
7 and 19 + 7 = 26, thereby creating a puzzle that was solvable in two substitutions.
From this point on, all the statements they entered were substitutive with respect
to the boxed expression. The final puzzle they made contained three statement
types and was solvable in four substitutions, as shown in Figure 6 (11 + 9 = 20
was superfluous in terms of solving the puzzle). This strongly suggests they were
able to emphasize and de-emphasize appropriately the substitutive and basic relational meanings of the equals sign.
Figure 6. A puzzle made by the children in Trial 5. It contains a variety of statement types
and can be solved in four substitutions.
Derek and John (Trial 6) made two distinct puzzles that were populated mainly
by statements that were basic relations (they were true) and equal to the boxed
Figure 7. A puzzle made by Derek and John in Trial 6. It contains a variety of statement
types, but most are equal to the value of the boxed expression and cannot be used to make
substitutions.
Ian Jones and Dave Pratt
21
expression, but were not substitutive with regard to it. Figure 7 shows their first
puzzle when it was about half made (they subsequently deleted many of the statements, and the final version was somewhat simpler). Only two of the statements
can be used to make a substitution (20 + 4 = 24 and 49 + 24 = 24 + 49), and the rest
are superfluous in terms of transforming the boxed expression into its single
numeral equivalent. They copied the superficial appearance of the puzzles
presented in Component 1 by using different statement types and clustering them
together. John was quite explicit about copying the format of the puzzles from
Component 1, saying, for example, that 40 + 33 = 73 needed to be at the bottom
because it was “the answer” (compare Figure 7 with Figure 3). The researcher asked
them which statements they thought could be used to solve the puzzle. This led
John to identify the need for statements that match the numerals in the boxed
expression: “We need to make like a 24 and 49 or something . . . 24 . . . and then
something in the sum which is 24.” Following this, he entered 20 + 4 = 24. The
researcher asked John to explain his reasoning to Derek, who said he did not understand what John was doing. However, John gave an answer not based on substitutive
effects but on his recall that some of the puzzles they had solved contained at least
two statements of the type c = a + b: “We need more than one of them, don’t we?
Just can’t have one there.”5 Later, with researcher prompting (“Is there anything
you can do about that?”), John did identify the need for the statement 32 + 16 = 16
+ 32 in order to commute 32 + 16 such that the statement 16 + 32 = 48 could be
used. However, this was the only clear articulation of a substitutive view during
Component 2 of Trial 6.
With one exception, John justified the few substitutive statements they did
Figure 8. A puzzle made by the children in Trial 7.
5 Note that John in fact entered his intended c = a + b in the form a + b = c. Such reversals were common throughout Study 3. This appears to be because the keypad tool for entering statements allowed
the user to begin on either the left or right side of the equals sign, as shown in Figure 2. Nevertheless,
the children’s indifference to left-to-right readings of statements is interesting.
22
A Substituting Meaning for the Equals Sign
construct in terms of superficial resemblance with puzzles they had previously
solved. They emphasized a basic relational meaning of the equals sign and only
rarely emphasized a substitutive meaning.
Colin and Imogen (Trial 7) produced four puzzles, although the first three
contained only one or two statements and were solvable in a single substitution.
The final puzzle contained 10 statements (Figure 8) and superficially resembled
the appearance of the puzzles from Component 1, but was again solvable in a single
substitution (149 = 55 + 94). Some of the statements were neither numerically equal
to the boxed expression nor substitutive with regard to it (e.g., 20 = 10 + 10). These
were willfully superfluous statements that Colin suggested would make the puzzle
more difficult (“We just need some sums that don’t do anything”). Some of the
statements were substitutive with regard to the boxed expression (e.g. 94 = 80 +
14) but were superfluous in terms of solving the puzzle.
Their discussion was most often focused on computation (“45 add 45 is equal to
. . .” and so on) and generating statements with a single numeral on one side of the
equals sign, suggesting that they often viewed the equals sign as a place indicator
for a result. There were exceptions, however, particularly toward the end of the trial.
At one point Colin discovered that 90 + 3 = 93 failed to make a substitution in 64 +
29 and suggested it might work if written as 93 = 90 + 3. Imogen disagreed, saying,
“It’s the same sum but just with the numbers swapped round. I know what will
work . . . put 64, add 29 there, and you put 93 there [i.e., enter the statement 64 + 29
= 93] and that works. Because that’s the sum.” Here it seemed that Colin was emphasizing a basic relational view because his focus was on the sameness of 90 + 3 and
93, and Imogen was emphasizing a substitutive meaning, realizing the need for
symbols to match exactly in order for a substitution to be made. This meaning does
not appear to have been stable, however, and for most of the remainder of the trial
Imogen mainly focused on computing results. Colin, conversely, increasingly
suggested statements in terms of their substitutive effects. Toward the end of the trial
Imogen was trying to input a statement that would transform 55 + 94, which was in
the boxed expression. Colin said, “You need a sum that equals 55 and equals 94.”
After some computational discussion Imogen entered 55 = 45 + 10 and 94 = 80 +
14. Colin said, “Split them up,” and Imogen used the statements to transform the
boxed expression from 55 + 94 into 45 + 10 + 80 + 14. This suggests Colin was
developing a stable substitutive meaning for the equals sign and was able to construct
statements in terms of required substitutive effects (partition, in this case). However,
this came late in the trial, which then ended—due to expiration of time—before the
children succeeded in making a puzzle that was solvable in several steps.
In sum, during Component 2 some children appropriately emphasized the basic
relational and substitutive meanings when making puzzles, but differences in
performance were clear. Only Bridie and Nadine (Trial 5) were able to complete
the task in the sense of producing solvable multistep puzzles, and their discussion
strongly suggests that they appropriately emphasized both meanings of the equals
sign. The children in Trial 6 produced a puzzle solvable in two substitutions, and
those in Trial 7 did not produce any solvable multistep puzzles. Moreover, the
Ian Jones and Dave Pratt
23
children in Trials 6 and 7 appeared to justify their work in terms of their puzzles
superficially resembling those from Component 1 of the trial. The data suggest that
these children viewed the equals sign substitutively at times, but struggled to coordinate it with the basic relational meaning.
Component 3: How can children connect their implicit arithmetical knowledge
with explicit transformations of notation? The final research question explored
whether the children could express their existing mental calculation strategies for
two-digit addition as solvable puzzles. For example, Qualifications and Curriculum
Agency (2001) gave an example of a child (Tony) who wrote his mental calculation
strategy for 30 + 41 as 30 + 40 = 70 and 70 + 1 = 71. We imagined such a child
entering these statements into the Sum Puzzles software and then being asked to
enter more statements such that it could be solved as a puzzle. Clearly Tony had
implicitly partitioned the 41 into 40 + 1 but had not written it as part of his strategy.
We were interested in whether children who are practiced in solving and making
puzzles could identify these implicit transformations and express them as explicit
substitutive statements. Were children to do this, it would suggest that they were
able to connect the arithmetic they already knew with the formal notation on the
computer screen.
Only data from Trials 5 and 6 informed the fourth research question because time
expired in Trial 7 after Component 2. In Trial 5 the researcher put 37 + 58 into the
box at the top of the screen. When prompted, Nadine gave the following strategy
for obtaining the result: “30 add 50 is 80. And 8 add 7 is 15. So you just have to
add those together and you get 95.” The researcher then asked the children to use
this strategy to make a puzzle. Nadine entered the first part of her strategy as 30 +
50 = 80. She then noticed an impasse (line 30) and attempted to articulate the need
to separate the tens and units (line 32). Bridie then explicitly stated the need for a
statement that would partition 58 into the sum of 50 and 8 (line 38).
30. Nadine: Oh, now I can’t do it.
31. R:
Why can’t you do it?
32. Nadine:It’s impossible because you won’t have them on their own unless you take
them out, but how do you take them out?
33. R:
Any ideas, Bridie?
34. Bridie: If [pause] you did 37 . . . [starts to enter a statement starting with 37]
35. Nadine: . . . add 58.
36. Bridie: No. Actually I wouldn’t. [deletes the 37]
37. R:
Show us what you would do, Bridie.
38. Bridie:I was going to do that and then, put that. [enters 37 + 50 = 87] It won’t
work. . . . Because first of all you need to divide 50 and the 8. Partition it.
Bridie then deleted 37 + 50 = 87 from the puzzle. She entered 50 + 8 = 58 and
used it to transform the boxed term 37 + 58 into 37 + 50 + 8. The children continued
working for another 7 minutes, resulting in a solvable multistep puzzle that accu-
24
A Substituting Meaning for the Equals Sign
Figure 9. Nadine’s strategy for solving 37 + 58 expressed as a puzzle.
rately reflected Nadine’s strategy (Figure 9).
Following this, they similarly went on to make a puzzle reflecting Bridie’s slightly
more complicated spoken strategy for 37 + 58: “I added 30 and 50, which was 80.
And then 5 plus 3 equals 8, so if you add that 3 to, um, the 87, because you have
37 and 50. . . . So that would be 87. If you add a 3 onto the 87, you have 90, and
then from that 8 you’d have 5 left, so it’s 95.” The puzzle reflecting this is shown in
Figure 10, and the connection between Bridie’s spoken strategy and the puzzle is
set out in Table 4.
In Trial 6, the researcher put 37 + 48 into the box at the top of the screen and
asked Derek for his strategy: “I just added the 30 and 40 together, added the 8 and
7 together.” The researcher asked the children to use Derek’s strategy to make a
puzzle, and with prompting they entered 70 = 30 + 40, 15 = 8 + 7 = 15, and 85 =
70 + 15. (This “backwards” formulation of the statements appears to have been due
to the design of the keypads, as noted in footnote 5.) The researcher asked whether
these statements could be used to make a substitution in 37 + 48. John replied “not
really, no” and entered 85 = 37 + 48. This resulted in a single-step puzzle (it could
be solved simply by substituting 85 for 37 + 48) that did not reflect Derek’s strategy.
The researcher then challenged the children to make a puzzle using John’s spoken
strategy for 37 + 48 (“30 add 40 equals 70 . . . 7 add 8. I just thought the 8 equals
Figure 10. Bridie’s strategy for solving 37 + 58 expressed as a puzzle.
25
Ian Jones and Dave Pratt
Table 4
The Connection Between Bridie’s Spoken Strategy for Solving 37 + 58 and the Resultant
Puzzle Shown in Figure 10
Spoken strategy
“I added 30 and 50, which was
80.”
Statements used
Resultant expression
30 + 7
=
37
30 + 7 + 58
30 + 7
=
7 + 30
7 + 30 + 58
50 + 8
=
58
7 + 30 + 50 + 8
30 + 50
=
80
7 + 80 + 8
“And then 5 plus 3 equals 8, so
if you add that 3 to, um, the 87,
because you have 37 and 50 . . .
. So that would be 87.”
80 + 7
=
7 + 80
80 + 7 + 8
5+3
=
8
80 + 7 + 5 + 3
5+3
=
3+5
80 + 7 + 3 + 5
“If you add a 3 onto the 87, you
have 90 . . . .” [87 is partitioned
as 80 + 7 and 90 is partitioned
as 80 + 10]
7+3
=
10
80 + 10 + 5
15
=
10 + 5
80 + 15
80 + 15
=
95
95
“…and then from that 8 you’d
have 5 left, so it’s 95.” [90
remains as
80 + 10, and 5 is added to 10,
not 90, giving 80 + 15 rather
than 90 + 5]
10, and then 7’s 17. 8 to 10 is 2, 17 takeaway 2 is 15. And 70 add 15 equals 85”).
They began by entering the compositions 70 = 30 + 40, 10 = 8 + 2, 17 = 10 + 7,
and 17 = 15 + 2. Following this, the researcher prompted them to think about
whether the statements could be used to make substitutions in the boxed expression.
Derek suggested a commuting statement (“We need one of them swap-arounds”),
although he gave no specific example or reasoning. The researcher prompted them
to consider the first expression they had entered, 30 + 40, and brought on screen
the puzzle they had made earlier (Figure 7). The researcher asked which statement
comes first when solving the puzzle in Figure 7. John described the partitioning
effect of using 20 + 4 = 24 to make a substitution of the numeral 24 (line 39), and
realized that an analogous statement was required but was unable to offer a clear
example (line 41).
39. John:That separates it.
40. R: Okay. So would something like that help us in our puzzle we’re doing now?
41. John:Oh yes! So we do like seven-, oh no, because they’re both tens. 30 add 0.
Following this, Derek claimed he knew what to do, and suggested and entered a
partitioning statement. However, his statement, 85 = 80 + 5, was numerically
equivalent to, rather than substitutive with regard to, the boxed expression. At this
point time ran out and the trial concluded.
26
A Substituting Meaning for the Equals Sign
The outcome of Component 3 was that only the children in Trial 5 succeeded in
implementing their spoken arithmetic strategies as solvable puzzles. They succeeded
because they were able to identify and make explicit the implicit transformations in
their spoken strategies, as is required by the task goal. This appears to have been a
continuation of their developing and increasingly coordinated meanings for the
equals sign through Components 1 and 2. In particular, they quickly identified the
need for a statement to partition the numerals in the given expression. Contrastingly,
the children in Trial 6 failed to implement their spoken strategies as solvable puzzles.
This appears to be because, unlike the children in Trial 5, they failed to make explicit
the partitions implied in their spoken calculation strategies, despite a clue from the
researcher. The children emphasized the substitutive meaning of the equals sign
during puzzle solving but only partially coordinated it with the basic relational
meaning when making puzzles. This partial coordination was not adequate for
making puzzles that were solvable in several steps (Component 2) or that reflected
the children’s spoken calculation strategies (Component 3).
DISCUSSION
We investigated the design affordances of computer-based tasks that can promote
a substitutive meaning for the equals sign in arithmetic notating tasks. We sought
evidence that when working with the Sum Puzzles software, children viewed and
talked about arithmetic statements in qualitatively different ways from those reported
in the literature. We now revisit the four research questions and discuss the evidence
presented previously in light of the diagrammatic framework set out earlier.
How Can the Substitutive Meaning for the Equals Sign
Promote Attention to Statement Structure?
The diagrammatic approach to task design involves enabling learners to explore
and experiment with formal mathematical representations. Central to diagrammatic
reasoning is predicting and observing the visual effects of making rule-governed
transformations. The first research question investigated whether the substitutive
meaning for the equals sign as implemented in the Sum Puzzles software can
support using equality statements as rules for making transformations of notation.
Analysis of the children’s discussion shows that they engaged with statement
structure in two key ways. First, they articulated visual searches for matches of
notation in order to identify where substitutions could be made. Such articulation
of visual searching was evident across all three trials to a greater or lesser degree,
as can be seen in Figure 5. This may seem somewhat trivial, but it stands in marked
contrast to the kinds of computational activities normally expected when children
are presented with formal arithmetic notation. Moreover, it provided a foundation
throughout all the trials for richer diagrammatic activities.
Second, the children articulated the distinctive visual substitutive effects
of different statement types. The substitutive effects of commutative statements
were typically articulated using phrases such as “changing round” and “switching,”
Ian Jones and Dave Pratt
27
and the substitutive effects of partitioning statements were typically described as
“splitting up” or “separating.” The articulation of the distinctive substitutive effects
of different statement types was evident across the children’s puzzle-solving
activities in all seven trials, as can be seen in Figure 5. More precisely, commutative
effects were articulated in all the trials, and partitioning effects to a lesser extent in
five of the trials.6 This diagrammatic approach to engaging children with the structural properties of arithmetic notation through observations of visual substitutive
effects is, to the best of our knowledge, novel.
Are the Substitutive and Basic Relational Meanings
for the Equals Sign Pedagogically Distinct?
In Study 1 the children considered arithmetic statements in terms of making
substitutions, but we suspected that they were unaware of the truth or falsity of the
statements. In Study 2 we explicitly tested whether they were indeed oblivious to
the basic relational meaning by including false equalities in some of the puzzles.
None of the children commented on their presence until they had been working on
the final puzzle for several minutes. At the conclusion of the trials we asked whether
they had noticed if any previous puzzles contained false equalities, and all the
children were unable to say. The children had therefore engaged in the activity of
making substitutions but had not concerned themselves with whether this was
mathematically sensible. This confirmed that the substitutive and basic relational
meanings for the equals sign are pedagogically distinct, and that puzzle solving can
emphasize the former over the latter.
How Can Children Appropriately Emphasize the
Substitutive and Basic Relational Meanings for the Equals Sign?
The task goal of solving presented puzzles emphasizes the substitutive meaning
and de-emphasizes the basic relational meaning. The task goal of making puzzles
instead emphasizes both meanings because when entering statements it is necessary
to ensure that they are true (the basic relational meaning) and, if the puzzle is to be
solvable, that they are substitutive with respect to the boxed expression. We tested
whether challenging children to make their own puzzles would enable them to
flexibly and appropriately emphasize both meanings.
The evidence sought was twofold. First, if the children were able to make solvable
multistep puzzles, this would suggest that they had appropriately emphasized both
meanings. We found that only the children in Trial 5, and to a limited extent the
children in Trial 6, managed to make solvable multistep puzzles, and that the children in Trial 7 did not manage to do so. Second, we analyzed the children’s discussion for the articulation of substitutive effects and numerical balance of statements.
We found that the children in Trial 5 viewed the equals sign as both a symbol of
6 Note that when puzzles contain commutative statements, it is not necessary—but it is useful—to
discern their distinctive commutative effects, and the children in Trial 4 did not do so while efficiently
solving the first 10 puzzles.
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A Substituting Meaning for the Equals Sign
substitution and basic relation, and that they were able to emphasize these two
meanings flexibly in order to achieve the task goal. One of the children in Trial 6,
John, viewed the equals sign in terms of both meanings at times, but the basic
relational view was dominant and his puzzle-making success was limited. In Trial
7, the children engaged substantially in calculating results in order to construct
numerically balanced statements. Toward the end of that trial Colin began to view
statements in terms of the substitutions they could make in the boxed expression,
but time expired before the children made a solvable puzzle.
The key finding here is that the altered task enabled some of the children to
appropriately emphasize both meanings of the equals sign toward achieving the
goal of making their own puzzles. However, the difference in performance across
the three trials was marked. The children in Trials 6 and 7 struggled to create solvable multistep puzzles, and those that they produced bore only a superficial resemblance to the puzzles they had solved in Component 1. They often focused on a
basic relational meaning, as evidenced by their use of expressions that were
numerically equal to the total of the boxed expression but could not be used to make
substitutions. As such, the substitutive meaning was not coordinated with the basic
relational meaning.
The puzzle-making activities of all the children, whatever their degree of success,
contrasts with the puzzle-solving activities reported above. In particular, Study 2
demonstrated that, when puzzle solving, children were engaged with making
substitutions toward a specified task goal but without consideration of the truthfulness of the presented statements. They were engaged and successful at solving
puzzles, but their activities were mathematically meaningless. In contrast, making
puzzles presented a greater challenge and necessitated consideration of the truthfulness and substitutive properties of the statements on the computer screen.
How Can Children Connect Their Implicit Arithmetical Knowledge
With Explicit Transformations of Notation?
The final research question was designed to determine whether children could
connect their existing knowledge of arithmetic with making puzzles. In particular,
we wanted to determine whether they could identify the transformations that are
commonly implicit in typical two-digit addition strategies and express them as
explicit arithmetic statements on the computer screen. Only the children in Trial 5
(and not those in Trials 6 or 7) managed to make solvable multistep puzzles that
accurately reflected their mental calculation strategies. The key to their success
appears to have been identifying the need to partition the numerals in the boxed
expression in order to be able to commute and transform that expression into a
single numeral. The children in Trial 6 did not construct a puzzle that reflected their
mental calculation strategies. This appears to have been due to their struggling to
partition the numerals in the boxed expression, despite nearly doing so and despite
prompts from the researcher. They viewed the equals sign in terms of basic relations
and substitution but seemed to be unable to coordinate these two views to produce
the statements required to complete their puzzles.
29
Ian Jones and Dave Pratt
LIMITATIONS TO THE STUDY
The three studies presented provided qualitative data to address the four research
questions, as discussed previously. However, the nature of the evidence presented
means that there are limitations to the type and strength of conclusions that can be
drawn.
The first limitation is the small number of participants used. In total, 14 students
participated across the seven trials reported. This enabled us to closely scrutinize
rich multimedia data in order to make inferences about the nature of the students’
mathematical activity and contrast this with findings in the literature. This was
appropriate given the innovative nature of the software-based task as an exploratory
window onto mathematical meanings (Noss & Hoyles, 1996). In particular, we were
able to demonstrate that the task encouraged the students to talk about and work
with arithmetic statements in novel and educationally promising ways. This
involved the development of a qualitative coding scheme as described in the paper.
However, the major drawback to this approach is that our conclusions cannot be
generalized in a statistical sense to a larger population of students. Such a generalization would require not merely a larger number of participants, but also the use
of standardized tests and measures that go beyond a specially developed qualitative
coding scheme.
Another limitation is that our data captured only how students talked about and
worked with arithmetic equations and did not measure learning or transfer.
Underlying the study is the theoretical assumption that the novel activities reported
have potential for mathematical learning. No claims are made that the students
learned during the trials, or that a single trial with the software, as experienced by
each participant, is substantial enough to foster conceptual change about the equals
sign or arithmetic statements. In order to test the particular software’s effectiveness
for learning, and for arithmetical substituting tasks in general, a larger number of
students undertaking a larger number of sessions would be required, along with the
administration of standardized tests and other measures.
A related limitation is the exclusive use of a digital medium for supporting the
substituting meaning of the equals sign. In principle, the substituting meaning could
also be supported by nondigital media such as card-sorting activities and worksheets. However, we are unable to say to what extent the approach presented is
applicable to such media.
Finally, we make no claims and provide no evidence about the applicability of the
substituting meaning of the equals sign to other mathematical domains. The puzzles
presented to learners exclusively contained arithmetic statements comprising only
positive integers and addition operations. Moreover, we limited the arithmetic principles embodied by particular statement examples to composition (a + b = c), commutation (a + b = b + a) and partitioning (c = a + b). There is also scope to support a
wider variety of statements that include negative numbers and decimal and fraction
representations, as well as other arithmetic operations, inclusion symbols, and so on.
This in turn would enable the embodiment of a much wider range of arithmetic
30
A Substituting Meaning for the Equals Sign
principles. More generally, the diagrammatic approach employed has scope for any
mathematical domain that involves reversible equivalence relations represented by
an equals sign. Further studies with modified versions of the software would be
required to explore the wider applicability of the approach.
CONCLUSION
We have presented qualitative evidence that a substitutive meaning for the equals
sign can meaningfully engage learners with the structure of arithmetic equality
statements. The computer-based tasks allowed children to predict and observe the
distinctive substitutive effects of different statement types. In particular, commutative statements were viewed in terms of their potential to “swap” numerals, and
partitioning statements were viewed in terms of their potential to “split” numerals
up. When pairs of children solved arithmetic puzzles comprising sets of equality
statements, they emphasized the substitutive meaning of the equals sign and
de-emphasized the basic relational meaning. The puzzle-solving task successfully
engaged the children in making substitutions, but they were focused on the task
goal of solving puzzles without consideration for mathematical coherence.
When children attempted to make their own puzzles, both meanings were evident
in their discussion. Only those children deemed mathematical high achievers by
their teacher were able to flexibly coordinate the two meanings as appropriate and
therefore successfully complete the task goals. These same children also demonstrated that they could connect their existing arithmetic knowledge with substitutive
statements on the computer screen by successfully constructing puzzles based on
their own mental calculation strategies. Conversely, children deemed mathematical
medium achievers switched between the two meanings, but not in a coordinated
manner. In particular, they were unable to identify and enter statements that would
have helpfully partitioned a given numeral into two numerals, and so were only
partially successful at making puzzles. Presenting to such children puzzles to solve
that contained repeated examples of statements that have a partitioning substitutive
effect may have assisted them.
These findings have implications for how researchers conceive of the “full”
relational conception of the equals sign, which comprises a basic relational view
(understanding both sides are the same) along with conceptual knowledge of the
structural properties of properly formed arithmetic notation. We propose that a
substitutive view of the equals sign is an important aspect of understanding the
structural properties of arithmetic statements. We have demonstrated one way in
which substitution allows access to thinking about statement structure in terms of
the transformative potential of commutative and partitioning statements.
Developmental models of children’s evolving meanings for the equals sign (e.g.,
Carpenter et al., 2003; Rittle-Johnson et al., 2011) start with the place-indicator
view and move via a basic relational view through to a full relational conception.
Our findings suggest there is scope to consider more explicitly how children’s
notions of substitution may play a part in this development. Substitution is clearly
important for understanding equivalence in algebraic contexts and is likely to be
31
Ian Jones and Dave Pratt
an aspect of the “didactic cut” (Filloy & Rojano, 1989), which requires learners to
substantially change how they think about and work with equations when they meet
algebra at the start of secondary schooling. For example, it is necessary to accept
unknowns on both sides of the equals sign, and this enables new actions such as
“substituting a numerical value for the unknown” (Filloy & Rojano, 1989, p. 19).
A better understanding of the role of substitution in children’s earlier development
of conceptions of the equals sign might aid our understanding of the difficulties
involved in moving from arithmetic to algebra. Moreover, Filloy et al. (2010) report
a further didactic cut: For some students, difficulties with substitution do not
emerge until the introduction of systems of simultaneous two-variable equations.
Substitution might then be an important aspect of a developmental model of the
evolution of meanings from the place-indicator view in young primary children
through to a fully operationalized understanding of equivalence in older secondary
students.
The findings also have implications for the teaching and learning of arithmetic
in primary and lower secondary school. Teaching the substitutive meaning to children may ease the transition from arithmetic to algebra as well as later transitions
to increasingly difficult algebra, such as the introduction of systems of simultaneous two-variable equations. The substitutive meaning also offers children access
to investigative and exploratory ways of thinking about and working with formal
notation, as described in this article. A particular challenge for teachers and
designers of educational materials is helping children to coordinate the substitutive
meaning with the basic relational meaning for the equals sign in arithmetic contexts.
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Authors
Ian Jones, Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU,
UK; I.Jones@lboro.ac.uk
Dave Pratt, Department of Geography, Enterprise, Mathematics and Science, Institute of Education,
University of London, WC1H 0AL; D.Pratt@ioe.ac.uk
Accepted August 16, 2011