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The present correlational study examined third- and fourth-graders’ (N = 56) knowledge of mathematical equivalence after classroom instruction on the equal sign. Three distinct learning trajectories of student equivalence knowledge were compared: those who did not learn from instruction (Never Solvers), those whose performance improved after instruction (Learners), and those who had strong performance before instruction and maintained it throughout the study (Solvers). Learners and Solvers performed similarly on measures of equivalence knowledge after instruction. Both groups demonstrated high retention rates and defined the equal sign relationally, regardless of whether they had learned how to solve equivalence problems before or during instruction. Never Solvers had relatively weak arithmetical (nonsymbolic) equivalence knowledge and provided operational definitions of the equal sign after instruction.

Long-term effects of learning algebra in high school include exposure to advanced mathematics courses, higher academic performance by the end of high school, and increased graduation rates in college (

One way that children’s interpretations of the equal sign can be expressed is through their performance on

Children perform poorly on non-canonical equivalence problems (for a review, see

Like conceptual knowledge of mathematics more generally, equal sign understanding involves the interconnection of different facets of knowledge (

By the age of four, children have little trouble establishing whether two sets of concrete objects, such as blocks, are quantitatively equivalent (

It is thus perhaps not surprising, then, that examinations of the effects of instructional interventions also point to the complexity of children’s learning. In the first place, not all students (in some cases fewer than half) respond to instruction in intended ways, despite statistically significant effects on the average (

Because our objective is to explore students’ knowledge in relation to how they respond to classroom instruction on the equal sign, we argue that an individual differences approach can be useful for our purposes.

Similar findings were reported in a study by

It has been well established that working memory and arithmetic fluency are important predictors of mathematical processing in a number of domains (

Previous research using an individual differences approach examined students’ knowledge after instruction as a function of instruction type (

The instruction was delivered in third- and fourth-grade classrooms by four teachers who participated in a larger professional development (PD) project on mathematical equivalence and relational thinking. During the PD, the teachers were exposed to the conceptual underpinnings of the equal sign, different ways to explain its meaning, and a variety of strategies for solving open-number sentences. After the PD, the teachers delivered one lesson on the meaning of the equal sign to their students.

We assessed students’ problem solving before and after the lesson, which allowed us to identify student groups according to how they responded to the instruction. Within eight weeks of the post-assessment, we again measured students’ equal sign understanding using a variety of measures, administered both in class and in individual interviews. Learning trajectories emerged at this point on the bases of their problem-solving performance assessed before instruction, at post-assessment, and on their problem solving within the eight-week period after the post-assessment. We then compared different aspects of the students’ equivalence knowledge as a function of their learning trajectories. In particular, we explored how students with different learning trajectories differed in terms of their equivalence knowledge other than problem solving, namely generating their own definitions for the equal sign, evaluating the definitions of others, justifying their own strategies for solving equivalence problems, and solving equivalence problems in nonsymbolic contexts. We also examined the performance of the children in the learning trajectories on non-equivalence tasks.

The study was part of a larger project (

The province of Québec publishes an income index and a socio-economic index for all public schools in the province (

In conducting this investigation, we complied with the American Psychological Association and local standards (i.e., principles of Canada’s Tri-Council Policy) related to the ethical treatment of the participants. Parental permissions were obtained, and each participant provided assent. Additionally, full ethical approval from the university, the school board, and the schools’ governing boards was granted.

The sequence of assessment activities and classroom instruction is presented in

Phase | Time of Year | Activity |
---|---|---|

1 | January – February | In-class assessment: Equivalence Problem Solving test |

2 | March – April | Classroom instruction on the meaning of the equal sign |

3 | April – May | In-class assessment: Equivalence Problem Solving test |

4 | May | In-class assessment: Fluency, Evaluating Definitions task |

5 | June | Interview 1: TONI-3, Numbers Reversed |

6 | June | Interview 2: Symbolic task, Generating Definitions task, Nonsymbolic task |

Before and after instruction (Phases 1 and 3), the participants completed a test of equivalence problem solving using the Equivalence Problem Solving test (

Between eight and 13 weeks after the pre-instruction equivalence problem-solving measure, the teachers delivered a single instructional lesson to their students (Phase 2). The lesson adhered to instructional principles validated in previous research on mathematical equivalence (

In their respective classrooms, the teachers began the lesson by presenting an equivalence problem (e.g., 3 + 1 + 1 = 3 + __) and saying, “the goal of a problem like this is to find a number that fits in the blank so that when you put together the numbers on the left side of the equal sign, you’ll have the same amount as when you put together the numbers on the right side of the equal sign” (see

When presenting all number sentences, the teachers followed the lesson plan’s instruction to use one color for the numbers and symbols on the left side of the equal sign and a different color for the right side of the equal sign. Three teachers used a third color to represent the equal sign. The fourth teacher used the same color for the equal sign as the color used for the left side of the equation. The lesson plan also required teachers to use sweeping hand gestures over the left and right sides of the equations. After the demonstrations, the teachers put additional problems on the board (two teachers presented two problems and the other teachers each presented three) and invited students to share their own strategies in a whole-class discussion. They then gave their students practice problems to work on in small groups or on their own. Three teachers assigned 15 practice problems and one teacher provided 8. The lengths of the lesson delivered by each teacher (excluding individual and small group practice) were 13.4 min, 22.8 min,^{1}

This teacher’s teaching style was more conversational than the other three teachers, which explains why the length of her lesson was relatively longer. Instructional fidelity data showed that the content covered in each of the teachers’ lessons was similar.

10.1 min, and 10.8 min (Eleven to 13 days after instruction, we administered the isomorphic version of the Equivalence Problem Solving test to the students in their classrooms (Phase 3) using the same procedures as before instruction. The post-instruction assessment took place in early April for one classroom and in early May for the other three classrooms. We compared students’ performance on the equivalence problem-solving measure before and after instruction to identify three initial learning trajectories: (1) students who performed poorly both before and after instruction; (2) those whose performance increased after instruction; and (3) those who performed well both before and after instruction.

Students’ knowledge and performance were subsequently assessed on three occasions. We first returned to the students’ classrooms between three and seven weeks after the post-instruction equivalence problem-solving measure (Phase 4). In this session we administered (a) a test of arithmetic fluency, and (b) a test in which students evaluated different definitions of the equal sign (Evaluating Definitions task). Second, within one week of the classroom visit, a member of the research team met individually with the students to administer measures of nonverbal intelligence (TONI-3) and working memory (Numbers Reversed; Phase 5). Third, no more than one week later, students met individually with an interviewer who assessed additional aspects of their equivalence knowledge (Phase 6): performance on an abbreviated version of the equivalence problem-solving measure (the Symbolic task), which served as a retention measure; justifications for their solutions on the Symbolic task; the quality of the definitions they provided for the equal sign (Generating Definitions task); and knowledge of arithmetical equivalence (the Nonsymbolic task). Students’ performance on the retention measure during the second interview was used to further differentiate the learning trajectories. All individual student meetings were videotaped, and the researcher was blind to the student’s group membership. The in-class assessments took place at the end of May, and the individual interviews took place in early June. For logistical reasons related to the larger project, we were not able to reduce the amount of time between the post-instruction classroom measures and the individual interviews.

The teachers delivered the equivalence lesson to their students in one mathematics class period. Each lesson was video recorded to enable assessment of instructional fidelity. We created a checklist that contained 13 essential lesson components (see

We administered three tasks to measure students’ non-equivalence skills: one measure of arithmetic fluency (^{rd} Edition (TONI-3;

The fluency measure (

The TONI-3 (

Instructions were given to the participant nonverbally, with gestures and facial expressions (e.g., pointing to test items; looking questioningly at the participant; shaking head, “no,” nodding head, “yes”; see the instruction manual in

Each student’s raw score was calculated by adding the number of correct responses between Item 1 and the ceiling item. The TONI-3 manual contains tables that were used to translate the raw scores into standardized scores, which were used in the analyses.

The third non-equivalence measure was the WJ-III Numbers Reversed (

Students received one point for each string of numbers they correctly repeated backward. The points earned were added to obtain a total score, which could range from 0 to 30.

Two equivalence assessments were conducted as a whole group with the students in their classrooms (the Equivalence Problem Solving test and the Evaluating Definitions task) and three were administered to students in one-on-one interview settings (i.e., the Symbolic task, the Generating Definitions task, and the Nonsymbolic task).

This test consisted of two- and three-term single-digit addition and subtraction problems. Students were asked to solve as many as they could by writing their answer on a blank line in the equation. The first five items were canonical addition and subtraction practice problems (e.g., 3 + 4 = ___). These were followed by 4 sets of 5 equivalence problems (e.g., 6 + 7 = ___ + 5), with 3 canonical problems interspersed between the sets and one as the last item on the test. This made for a total of 9 canonical problems and 20 non-canonical problems.

Five types of equivalence problems were used: (a) identity (

Accuracy (percent correct) was calculated separately for canonical and non-canonical problems. For the canonical problems, Cronbach’s alpha reliability estimate for the sample at pretest was .67 and .71 at posttest. These low estimates are likely due to the small number of canonical items and the near-ceiling performance on these items (

As a retention measure, students were asked to solve five equivalence problems similar to those on the Equivalence Problem Solving test. Each problem was presented on an individual index card, and students wrote their answers directly on the card. Accuracy (percent correct) was calculated to obtain an Equation Solving score. Reliability for the Equation Solving subscale of the Symbolic task was high, Cronbach’s α = .98.

After each item, the researcher asked the students to justify their answers. Each justification was coded as either relational or non-relational. Justifications coded as relational were either descriptions of the equal sign as relational (e.g., for 3 + 4 = 4 + __, “the left side has to equal the same as the right side. The left side is 7, and if I put a 3 on the line, the right side is going to equal 7, too.”) or descriptions of a procedure that was consistent with a relational view (e.g., “I added 3 and 4 and got 7, then I subtracted 4 from it and put 3 on the line.”). Justifications that were coded as non-relational included descriptions that were operational (e.g., for 6 + 4 + 5 = 6 + ___, “you always put the answer after the equal sign and so the answer is 15”), that involved a procedure that aligned with an operational view (e.g., “The answer is 15 because 6 plus 4 plus 5 is 15”), or that featured other non-relational views (e.g., “I just put any number on the line.”).

The student was awarded 1 point for each relational justification and 0 points for justifications coded as non-relational. The points were summed across all 5 items to yield a Justification score ranging from 0 to 5. Reliability for the Justification subscale was high, Cronbach’s α = .96. The first author coded all the justifications. A trained rater was asked to code 10% of the justifications, and an inter-rater agreement of 96% was obtained.

The researcher asked the students to generate their own definition of the equal sign. The child was presented with an index card on which the equation 3 + 4 = 2 + 5 was written. The researcher then pointed to the equal sign in the equation and asked, “What is the name of this symbol? Can you explain to me what this symbol means?”

Students’ definitions were coded as Relational, Operational, or Mixed. The interviewers were instructed to prompt students with follow-up questions whenever they provided answers that were difficult to classify (e.g., “It means equal.”). Prompting stopped when the interviewer felt she had determined the students’ interpretations of the equal sign. The definitions were coded as Relational when students explained that both sides of the equal sign needed to represent the same amount. Definitions were coded as Operational when descriptions contained a misconception, such as “add all the numbers” (e.g., answering 17 to solve the problem 8 + 4 = __ + 5) or “the answer comes next” (e.g., answering 12 to solve the problem 8 + 4 = __ + 5). In some cases, students provided both relational and operational aspects in their definitions, such as “The equal sign always means different things. Sometimes it means ‘the same as’ [relational], and sometimes it means you need to say the answer of the problem [operational].” We decided to place such definitions in a separate category, Mixed, because of how such responses could be placed in the construct map presented by

In this task (

Two definitions reflected a relational understanding of the equal sign (e.g., “both sides of the equal sign should have the same amount”), two reflected an operational understanding of the equal sign (e.g., “the answer goes next”), and two were not related to equivalence (e.g., “all the numbers after it are small”). Using the same scoring procedure as

The Nonsymbolic task (

The child was instructed to put blocks on the empty card “so that when you put together these on this side of the blue tent (interviewer gestured to the blocks on the left), you’ll have the same number as when you put together these on this side of the blue tent” (interviewer gestured to the blocks and the empty card on the right;

Means and standard deviations of equivalence problem-solving scores at pre-instruction and post-instruction by grade and by equation type (canonical, non-canonical) are presented in

Measures | Grade 3 |
Grade 4 |
||
---|---|---|---|---|

Pre-instruction | ||||

Canonical Problems | 81.94 | 21.36 | 90.00 | 12.24 |

Non-Canonical Problems | 28.98 | 34.35 | 25.12 | 41.43 |

Post-instruction | ||||

Canonical Problems | 87.61 | 17.86 | 92.22 | 14.71 |

Non-Canonical Problems | 69.89 | 39.50 | 71.17 | 38.22 |

Performance was analyzed with a 2(Grade: 3, 4) by 2(Time: pre-instruction, post-instruction) by 2(Equation Type: canonical, non-canonical) ANOVA with repeated measures on the last two variables. Mean performance was higher after instruction compared to before instruction,

Only non-canonical items were included in subsequent analyses. The length of time between the pretest and the posttest was not correlated with children’s difference scores on the non-canonical items (

Students’ learning trajectories were identified using their performance on the Equivalence Problem Solving test at pre- and post-instruction and refined using their performance on the Equation Solving component of the Symbolic task (i.e., the retention measure) administered during the interview at Phase 6 (see

Using the 60% criterion, three initial learning trajectories were formed: Nonsolvers (

We used the same 60% threshold for the students’ performance on the retention measure at Phase 6 to further refine the students’ learning trajectories. Thirteen of the 18 Nonsolvers performed below the 60% cutoff on the Symbolic task, meaning their performance was comparable to that on the post-instruction Equivalence Problem Solving test. The remaining five Nonsolvers learned how to solve non-canonical problems in the time between the posttest and the individual interview with the researcher; these students were classified as “Eventual Learners.” Of the 24 Learners, three students did not maintain their performance at 60% or higher after the posttest; we called these three students “Forgetters.” All 14 Solvers showed above-criterion accuracy during the interview.

Separating the Eventual Learners from the Nonsolvers group and the Forgetters from the Learners, we obtained the final learning trajectories, as shown in ^{2}(2,

A two-way mixed ANOVA was performed with teacher (4) as the between-subjects factor and time (3: pretest, posttest, retention measure) as the within-subjects factor, and percent correct as the dependent variable. There were main effects of teacher, ^{2} = .38, and time, ^{2} = .48. Post-hoc analyses using Bonferroni corrections revealed that mean scores were higher on the posttest than on the pretest and that they were also higher on the Symbolic task than on the pretest, both

To provide a richer picture of the students within each learning trajectory, we examined the strategies used by Never Solvers, Learners, and Solvers on the non-canonical problems on the Equivalence Problem Solving test at both pretest and posttest. Each student was placed in one of five categories at pretest based on the strategies they used to solve the problems. Students who used correct strategies (i.e., leading to the correct answer) on 80% of the items were placed in the Correct category. The remaining students used incorrect strategies on more than 20% of the items. These students were placed in the Add All the Numbers category if they used this strategy on at least 80% of the items and in the Answer Comes Next category if at least 80% of their strategies were of this type. If 80% or more of strategies were not discernable from the answers, the students were placed in the Other category. Finally, students whose strategies did not fall predominantly in any one of the above categories were placed in the Mixed category. We observed that, at pretest, Never Solvers were the only ones to be classified as Add All the Numbers, and half of the Never Solvers (6/13) were placed in either the Add All the Numbers or Answer Comes Next categories. Most of the Learners (17/21) were placed in the Mixed category, however. Finally, and as would be expected, most of the Solvers (12/14) were in the Correct category and thus used appropriate strategies to solve the items on the test. Two of the Solvers (2/14) were placed in the Mixed category.

We also looked at the change of strategies on the Equivalence Problem Solving test from pretest to posttest for each student. We found that of the 17 students who used a single arithmetic-based strategy at pretest (i.e., students who were placed either in the Add All the Numbers or Answer Comes Next categories), six (35%) moved to the Correct category at posttest – i.e., used predominantly correct strategies after instruction. Furthermore, of the 27 students who used a variety of strategies at pretest (i.e., students who were placed in the Mixed category), 18 (67%) used correct strategies at posttest. Most of the students who were not placed in the Correct category at pretest changed strategy categories from pretest to posttest; only two students who used a single arithmetic-based strategy remained in the same strategy category at both time points.

In this section, we report differences between the learning trajectories on the non-equivalence tasks and equivalence knowledge measures. Means and standard deviations of the scores on the non-equivalence tasks and the equivalence knowledge measures across grades as a function of learning trajectory can be found in

Measures | Solvers |
Learners |
Never Solvers |
Eventual Learners |
Forgetters |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Non-equivalence measures | |||||||||||||||

Fluency^{a} |
9.3 | 4.8 | 14 | 7.4 | 4.1 | 20 | 2.8 | 1.1 | 13 | 5.6 | 3.0 | 5 | 3.4 | 1.0 | 3 |

TONI-3 | 103.7 | 6.8 | 13 | 105.4 | 10.2 | 20 | 91.5 | 12.3 | 12 | 93.2 | 6.1 | 5 | 112.7 | 6.8 | 3 |

Numbers Reversed^{b} |
12.0 | 2.9 | 14 | 11.9 | 2.2 | 21 | 7.6 | 4.6 | 13 | 8.5 | 3.1 | 4 | 9.7 | 1.5 | 3 |

Equivalence knowledge | |||||||||||||||

Equation Solving^{c} |
97.1 | 7.3 | 14 | 100.0 | 0.0 | 21 | 3.1 | 11.1 | 13 | 88.0 | 17.9 | 5 | 0.0 | 0.0 | 3 |

Justification^{d} |
4.9 | 0.4 | 14 | 4.8 | 0.5 | 21 | 0.1 | 0.3 | 13 | 4.0 | 1.2 | 5 | 0.0 | 0.0 | 3 |

Evaluating Definitions^{e} |
0.1 | 0.8 | 14 | -0.5 | 0.9 | 21 | -0.5 | 0.8 | 12 | 0.3 | 0.8 | 5 | 0.0 | 0.9 | 3 |

Nonsymbolic^{c} |
100.0 | 0.0 | 14 | 93.3 | 14.6 | 21 | 70.9 | 35.1 | 11 | 92.0 | 17.9 | 5 | 93.3 | 11.5 | 3 |

^{a}min: 0, max: 26. ^{b}min: 0, max: 30. ^{c}Reported in percent. ^{d}min: 0, max: 5. ^{e}min: -2, max: 2.

A one-way ANOVA with Fluency as the dependent measure revealed differences among the Never Solvers, Learners, and Solvers, ^{2} = .32. Post-hoc tests with Bonferroni corrections indicated that the Never Solvers scored lower than both the Learners (^{2} = .27, and post-hoc tests with Bonferroni corrections again placed the Never Solvers lower on this measure than the Learners (^{2} = .28, and the post-hoc tests revealed again that the Never Solvers scored lower on this measure than the other two groups (Learners,

On the Justification component of the Symbolic task, Learners (

Students generated three types of definitions on the Generating Definitions task: operational, mixed (both relational and operational), and relational. The proportions of students in each of the five learning trajectories who provided these three definitions are presented in

Definition Type | Learning Trajectory |
||||
---|---|---|---|---|---|

Never Solvers | Learners | Solvers | Eventual Learners | Forgetters | |

Operational | 11 (84.6%) | 6 (28.6%) | 2 (14.3%) | 2 (40.0%) | 1 (33.3%) |

Mixed | 2 (15.4%) | 11 (52.4%) | 6 (42.9%) | 1 (20.0%) | 1 (33.3%) |

Relational | 0 (0.0%) | 4 (19.0%) | 6 (42.9%) | 2 (40.0%) | 1 (33.3%) |

Total | 13 | 21 | 14 | 5 | 3 |

To assess the relation between learning trajectory and definition type, the mixed and relational categories were collapsed into an “ever relational” category to represent definitions that contained at least some relational elements. A chi-square test of association was conducted between definition type (ever relational, operational) and learning trajectory (Never Solvers, Learners, and Solvers). A statistically significant association was found between definition type and learning trajectory, χ^{2}(2) = 15.84,

We conducted an ANOVA to test for group differences on the Evaluating Definitions task. Given that none of the non-equivalence measures was correlated with the Evaluating Definitions task, all

An ANOVA was conducted to test for group differences on the Nonsymbolic task. Although performance on each of the TONI-3 and Numbers Reversed was correlated with the Nonsymbolic task (^{2} = .25. Follow-up comparisons with Bonferroni corrections showed that the Learners and the Solvers outperformed the Never Solvers (

A considerable number of students do not respond to instruction about the equal sign in intended ways, but little is known about the nature of their equivalence knowledge relative to the ways in which they respond to classroom lessons on the equal sign. We contend that characterizing students’ knowledge as a function of how they respond to classroom instruction lends the ecological validity necessary for conclusions that are ultimately useful to practicing teachers. The objective of the present study, therefore, was to examine differences in the equivalence knowledge of students who responded in different ways to classroom instruction on the equal sign. Three main learning trajectories emerged from the data: those who performed poorly on a test of equation solving both before and after instruction, those who had improved, and those who performed well on the same test both before and after instruction. More tentatively, our data also revealed two additional trajectories, those who forgot what they had learned following instruction and those who at some point improved their problem-solving performance, despite not showing improvement immediately after instruction.

Students’ performance at posttest was consistent with previous literature showing how children’s misconceptions are resistant to change (e.g.,

When considering the strategies students used to solve equivalence problems at pretest, we found that, consistent with previous research, students who relied on one incorrect arithmetic-based strategy were less likely to solve the equivalence problems correctly at posttest (

We also found that students who failed to solve equivalence problems at both time points after instruction (i.e., the Never Solvers) had little in common with those whose problem solving improved at some point after instruction began, at least on the equivalence measures used in this study. Specifically, relative to students with different learning trajectories, the Never Solvers still struggled to solve equivalence problems, had relatively weak arithmetical (i.e., nonsymbolic) equivalence knowledge, and provided predominantly operational definitions of the equal sign immediately and several weeks after having received instruction. We observed a contrasting pattern for students in the other two primary learning trajectories, regardless of whether they knew how to solve equivalence problems before instruction (Solvers) or whether they showed improved performance afterward (Learners): Most retained their ability to solve problems, defined the equal sign relationally, and had almost no deficiencies in their arithmetical equivalence knowledge.

Despite students in the Never Solvers trajectory exhibiting such differences relative to students in the other four trajectories, more consistent findings were observed on the Nonsymbolic task, which revealed performance above 60% in all groups. Nevertheless, the Never Solvers still demonstrated relatively greater difficulty than their peers in the other four trajectories on the Nonsymbolic task. Additional research that focuses on the reasons that this group of students had greater difficulty than others is necessary for teachers to know how to respond appropriately during instruction. Showing similar consistency across trajectories, but in the opposite direction, all groups appeared to struggle when asked to evaluate others’ definitions of the equal sign.

In contrast to some previous studies (e.g.,

It is more difficult to explain the performance pattern for the Eventual Learners, the students who did not learn immediately after instruction but who performed above our threshold on problem solving a few weeks later. However these students eventually learned how to solve equivalence problems, their learning was accompanied by the ability to define the equal sign relationally and an understanding of arithmetical equivalence (i.e., high performance on nonsymbolic problems). One explanation for this finding is that once students learn how to solve equivalence problems, whether it is immediately after instruction or not, and whether it is retained over time, they are “primed” (e.g.,

Our results contribute to the literature by examining the nature of children’s equivalence knowledge, and their performance on non-equivalence tasks, in response to instruction on the equal sign. We found three primary trajectories – those who performed poorly before and after instruction, those who improved, and those who performed at a high level before and after instruction. The current research contributes to the literature by characterizing the nature of the equivalence and non-equivalence knowledge of the students in these three trajectories. Students who failed to perform well on equivalence problems after instruction and still struggled several weeks later showed generally weaker equivalence knowledge relative to those who showed improvement immediately after instruction, regardless of their problem-solving performance several weeks later. Another contribution of our study is that we provide suggestive evidence that there may be, in fact, two additional learning paths, those who forget and those who eventually learn. Even those students who forget what they have learned have stronger equivalence knowledge than those students who fail on equivalence problems at all time points. The robustness of these additional two trajectories, however, should be tested in future research.

The results of the present study contribute to existing literature on the relation between mathematics instruction and student learning. Theoretical and anecdotal accounts of teaching and learning in mathematics (

Certain limitations of our work should be noted. First, the number of students in each learning trajectory was small in some cases, preventing us from including other variables, such as gender, in our analyses. Our sample was also too small to help us arrive at reliable conclusions about the two smallest learning trajectories we identified (i.e., Eventual Learners and Forgetters).

Additionally, we were not able to document the instructional activities the teachers implemented in their classrooms other than the lesson that we had asked them to deliver. It is possible, for example, that some of them may have brought out the algebraic character of arithmetic in a number of other lessons (

The results of the present study are informative for teachers. Students who demonstrate persistent difficulties with the equal sign are likely the ones who struggle with many, if not most, aspects of equivalence knowledge. In the context of the classroom, then, students’ problem-solving performance may be a useful index for their overall equivalence knowledge. A key pedagogical implication is that students who have difficulty responding to lessons on the meaning of the equal sign may benefit from additional targeted instruction on equivalence in both symbolic and nonsymbolic contexts.

Furthermore, the added observation that students’ justifications on the retention measure revealed views of the equal sign that were consistent with their responses serves to support the validity of using problem-solving performance as an indicator of overall knowledge. The conclusion that problem solving is particularly revealing of students’ knowledge may only hold given the specific type of instruction delivered in the present study, however, which was focused on explicit explanations of the meaning of the equal sign and clear demonstrations on how to solve equivalence problems. Regardless, given that instruction served to constrain students’ learning about equivalence suggests that frequently assessing the aspects of equivalence that are highlighted during instruction would help to identify those students with persistent difficulties.

Finally, our research provides indirect evidence that the instruction designed by

Wrote “=” on the board and asked students what the symbol was

Wrote equivalence problem on the board

Used different colors to differentiate both sides

Explained to students the goal of equivalence problems

Gestured to each side during the lesson

Emphasized that there are two sides to an equivalence problem

Explained the meaning of “=”

Solved equivalence problem on the board

Gave extra examples

Showed exercises with missing equal sign

Gave students practice with missing equal sign problems

Gave students time to practice in small groups or pairs

Wrap-up with reiteration of what “=” means

The current research was supported by the Social Sciences and Humanities Research Council of Canada Grant 410-2009-0880.

The authors have declared that no competing interests exist.

Rebecca Watchorn Kong is now at Research & Experimentation, Financial Consumer Agency of Canada, Ottawa, ON, Canada. Jody Sherman LeVos is now at HOMER by BEGiN, New York, NY, United States.

The authors have no additional (i.e., non-financial) support to report.