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Using Anticipatory Diagrammatic Self-explanation to Support

Learning and Performance in Early Algebra

Tomohiro Nagashima, Carnegie Mellon University, tnagashi@cs.cmu.edu

Anna N. Bartel, University of Wisconsin, Madison, anbartel@wisc.edu

Gautam Yadav, Carnegie Mellon University, gyadav@andrew.cmu.edu

Stephanie Tseng, Carnegie Mellon University, stseng2@andrew.cmu.edu

Nicholas A. Vest, University of Wisconsin, Madison, navest@wisc.edu

Elena M. Silla, University of Wisconsin, Madison, esilla@wisc.edu

Martha W. Alibali, University of Wisconsin, Madison, martha.alibali@wisc.edu

Vincent Aleven, Carnegie Mellon University, aleven@cs.cmu.edu

Abstract: Prior research shows that self-explanation promotes understanding by helping

learners connect new knowledge with prior knowledge. However, despite ample evidence

supporting the effectiveness of self-explanation, an instructional design challenge emerges in

how best to scaffold self-explanation. In particular, it is an open challenge to design self-

explanation support that simultaneously facilitates performance and learning outcomes.

Towards this goal, we designed anticipatory diagrammatic self-explanation, a novel form of

self-explanation embedded in an Intelligent Tutoring System (ITS). In our ITS, anticipatory

diagrammatic self-explanation scaffolds learners by providing visual representations to help

learners predict an upcoming strategic step in algebra problem solving. A classroom experiment

with 108 middle-school students found that anticipatory diagrammatic self-explanation helped

students learn formal algebraic strategies and significantly improve their problem-solving

performance. This study contributes to understanding of how self-explanation can be scaffolded

to support learning and performance.

Introduction

Self-explanation

Self-explanation is a learning strategy in which learners attempt to make sense of what they learn by generating

explanations to themselves (Chi et al.,1989; Rittle-Johnson et al., 2017). A number of studies have provided

evidence for the effectiveness of self-explanation across domains (Ainsworth & Loizou, 2003; Bisra et al., 2018).

From a cognitive perspective, self-explanation helps learners integrate to-be-learned information with their prior

knowledge, leading to deeper understanding of the content (Bisra et al., 2018). For example, in the context of

problem solving in mathematics, learners may be asked to provide reasoning for their solved steps in order to

deepen their conceptual understanding of the procedures. Although self-explanation activities may take different

forms (e.g., explaining worked examples, explaining while solving problems, and explaining text passages), they

share the core principle of supporting deeper understanding through connecting new content with existing

knowledge.

Scaffolding self-explanation as a challenging design problem

The demonstrated effectiveness of self-explanation does not guarantee that effective self-explanation activities

are easily designed. Self-explanation can be a demanding task for learners. It has been reported that scaffolding

self-explanation activities facilitates learning (Rittle-Johnson et al., 2017). Prior studies have designed and tested

various types of scaffolded self-explanation, such as presenting menu-based, multiple-choice explanations

(Aleven & Koedinger, 2002; Berthold et al., 2011; Rau et al., 2015), providing training on self-explanation (Hodds

et al., 2014), using visual representations (Ainsworth & Loizou, 2003; Nagashima, Bartel et al., 2020), using

contrasting cases (Sidney et al., 2015), and providing feedback on self-explanation (Aleven & Koedinger, 2002).

All of these types of self-explanation support have been shown to be effective. Yet, there are still

challenges in how best to design optimal scaffolding support for self-explanation. A first challenge lies in how to

design scaffolded self-explanation to promote both conceptual and procedural knowledge. Acquiring both

conceptual and procedural knowledge is fundamental to learning (Rittle-Johnson & Alibali, 1999); however,

studies on scaffolded self-explanation have typically shown it to be effective for enhancing either conceptual

knowledge or procedural knowledge, but not both (Berthold et al., 2011; Nagashima, Bartel et al., 2020; Rau et

al., 2015, but see Aleven & Koedinger, 2002). Rittle-Johnson et al. (2017) explain that this disassociation may be

due to the unique characteristics of specific forms of scaffolding. Self-explanation scaffolding designed to focus

on one aspect of content may hinder learners’ focus on other important aspects. For example, asking students to

select a correct conceptual explanation from among a list of similar explanations in a multiple-choice format

would encourage learners to focus on conceptual understanding of the content, but it would not give an opportunity

for learners to develop their procedural skills (e.g., problem-solving skills).

A second challenge is how to design scaffolded self-explanation that enhances problem-solving

performance when combined with, or embedded in, problem-solving activities. Self-explanation can be time-

consuming, and because self-explanation requires learners to engage in additional cognitive activities, learners

who receive self-explanation support may solve fewer problems in a limited amount of time compared to solving

problems without self-explanation support. If scaffolded appropriately during self-explanation, learners’

performance on the target task would improve. This would result in efficient learning (i.e., learners with self-

explanation achieve similar learning gains with fewer problems or less time spent compared to those without self-

explanation). Most prior studies of self-explanation do not report measures of the problem-solving performance

and efficiency of learning with self-explanation, such as time spent on the task (Bisra et al., 2018; but see Aleven

& Koedinger, 2002). In sum, there are persistent design challenges in how to design effective and efficient self-

explanation that supports both learning and performance.

Designing evidence-based self-explanation scaffolding

To approach these challenges, we designed self-explanation support for a web-based educational software called

an Intelligent Tutoring System (ITS) for algebra problem solving (Long & Aleven, 2014). In our design, self-

explanation is interleaved with problem solving; learners are asked to explain the next strategic problem-solving

step in the form of a diagram before doing the same step in symbols (Figures 1-3). They receive feedback from

the ITS both on their explanation and their step using mathematical symbols. We designed the self-explanation

support following several evidence-based principles from cognitive psychology, educational psychology,

instructional design, and the learning sciences, which we describe below.

Figure 1. The ITS starts by asking a learner to select a correct diagram for the given equation. The ITS gives

correctness feedback on the learner’s choice of diagram.

Figure 2. Next, the ITS asks the learner to explain (by selecting a diagram) what would be a correct and

strategic step to take next. The ITS gives feedback on the choice of diagram.

Figure 3. After selecting a correct and strategic step, the learner enters the step in symbols.

Visual representations designed to support students’ conceptual understanding

Research has shown that visual representations can support conceptual understanding (Rau, 2017). Visual

representations can depict information that is difficult to express through verbal means and can make important

information salient. We chose a visual representation called tape diagrams, which are commonly used in algebra

classrooms in countries such as Japan, Singapore, and the United States (Booth & Koedinger, 2012; Chu et al.,

2017; Murata, 2008). Prior studies using tape diagrams in algebra problem solving show that tape diagrams help

students gain conceptual understanding and avoid conceptual errors (Chu et al., 2017; Nagashima, Bartel et al.,

2020). In particular, our own prior experiment found that diagrammatic self-explanation (in which students, after

each equation-solving step, are asked to select, from three options, a diagram that corresponds to the step) helped

learners gain conceptual knowledge in algebra (Nagashima, Bartel et al., 2020). In the present study, students are

similarly asked to choose tape diagrams as a way to explain their steps, following the principle of anticipatory

self-explanation (Bisra et al., 2018; Renkl, 1997), as explained next.

Anticipatory self-explanation to support understanding of problem-solving strategies

Anticipatory self-explanation is a type of self-explanation in which learners generate inferences about future steps.

Previously, Renkl (1997) found that, when prompted to talk aloud while studying worked examples that provided

solutions step-by-step, many successful self-explainers predicted solutions in advance. In algebra problem

solving, such anticipatory self-explanation, rather than post-hoc self-explanation, can potentially support

inference generation about strategic problem-solving steps (e.g., “what would be a good next step for the equation,

3x + 2 = 8?”). If students consider the mathematical symbols as the target representation to learn, engaging in

step-level anticipatory self-explanation could help students understand strategic next steps, which would improve

both understanding of strategic solution steps and problem-solving performance. On the other hand, post-hoc self-

explanation might not be particularly effective for helping students take strategic problem-solving steps.

Contrasting cases that differ on conceptual features and problem-solving strategies

The use of contrasting cases is an established instructional strategy in which learners are presented with

contrasting examples that differ in meaningful conceptual aspects (Schwartz et al., 2011). Contrasting cases help

learners notice meaningful differences. This instructional strategy is typically used with prompts for self-

explanation, to encourage learners to cognitively and constructively engage with the cases (Sidney et al., 2015).

In the self-explanation support used in the current study, three options of tape diagrams are displayed,

which differ in one conceptual aspect and one strategy-related aspect. For example, in Figure 2 the tutor displays

three diagrams that represent a correct and strategic next step (diagram on the left), an incorrect option (diagram

on the right, in which the subtraction is done on only one side of the equation) and an option that is correct but

not strategic (diagram in the middle, in which 2x was added to both sides, which does not get the learner closer to

the solution). This set of options allows learners to distinguish, not only between correct and incorrect steps, but

also between correct and strategic steps and correct but not strategic steps. In problem states in which two correct

and strategic steps are available (e.g., subtracting 2x from 8x = 2x + 6 or dividing both sides by 2), the ITS shows

those two options and one incorrect option. Engaging with contrasting cases prior to practicing the target problem-

solving skill with symbols might be particularly meaningful, because students would be able to follow the selected

diagram option when entering the solution step with symbols and thereby learn to use correct and strategic steps.

Present investigation and hypotheses

In the present study, we investigate the effectiveness of scaffolded self-explanation support on learning and

performance. We hypothesize that (H1) the anticipatory diagrammatic self-explanation will promote students’

conceptual understanding, enhance procedural skills, and help students learn formal algebraic strategies. We also

hypothesize that (H2) the anticipatory diagrammatic self-explanation will enhance performance during problem

solving in the ITS; students with the support will perform better on learning process measures (e.g., fewer hint

requests and fewer incorrect attempts per step) while solving symbolic problem-solving steps, and they will solve

a similar number of problems as students who do not receive the scaffolded self-explanation support.

Method

Participants

We conducted an in vivo experiment (i.e., a randomized controlled experiment in a real classroom context) at two

private schools in the United States. Participants included 55 6th graders and 54 7th graders across nine class

sections taught by four teachers. The experiment was conducted in October 2020, when both schools adopted a

hybrid teaching mode in which the majority of students (n = 102) attended study sessions in-person and the rest

of the students attended remotely (n = 7). Teachers reported that they had never focused their instruction on tape

diagrams, although they indicated that some students might have seen tape diagrams in their learning materials.

Materials

Intelligent Tutoring System for equation solving

In addition to the anticipatory diagrammatic self-explanation ITS described above, we used a version that did not

include tape diagrams (Figure 4) (Long & Aleven, 2014). In this No-Diagram ITS, students learn to solve

equations step-by-step, but without diagrammatic self-explanation steps. All other features (e.g., step-level

feedback messages and hints) are the same as in the version with tape diagrams. Both ITS versions had four

different types of equations, which were chosen in consultation with the teachers (Table 1). We only used

equations with positive numbers since tape diagrams were not found useful for representing negative numbers

(Nagashima, Yang et al., 2020). Most of the participants in this study, per teachers’ report, had seen or practiced

Levels 1 and 2 problems, but had not learned Levels 3 and 4 problems.

Figure 4. A version of ITS with no diagrammatic self-explanation.

Table 1: Types of equations the tutor contained and the number of problems in the tutor

Equation type

Example

Number of problems in the ITS

Level 1

x + a = b

x + 3 = 5

4

Level 2

ax + b = c

2x + 3 = 7

5

Level 3

ax + b = cx

5x = 3x + 2

6

Level 4

ax + b = cx + d

5x +2 = 3x + 8

13

Test instruments

We developed web-based pretest and posttest assessments to assess students’ conceptual and procedural

knowledge of basic algebra. The tests contained several items drawn from our previous work (Nagashima, Bartel

et al., 2020) as well as new items. The conceptual knowledge items consisted of eight multiple-choice questions

and one open-ended question, which assessed a wide range of conceptual knowledge constructs, including like

terms, inverse operations, isolating variables, and the concept of keeping both sides of an equation equal. We also

included four problem-solving items (e.g., “solve for x: 3x + 2 = 8”), including two items that were similar to

those included in the ITS and two transfer items involving negative numbers. We developed two isomorphic

versions of the test that varied only with respect to the specific numbers used in the items; participants received

one form as pretest and the other as posttest (with versions counterbalanced across subjects).

Procedure

The study took place during two regular mathematics classes. The classes were virtually connected to the

experimenters and remote learners through a video conferencing system. Students were randomly assigned to

either the Diagram condition or the No-Diagram condition. In the Diagram condition, students used the ITS with

anticipatory diagrammatic self-explanation. In the No-Diagram condition, students used the ITS with no self-

explanation support. The only difference between the Diagram and No-Diagram conditions was whether students

self-explained their solution steps in the form of tape diagrams or not.

On the first day, all students first worked on the web-based pretest for 15 minutes. Then a teacher or the

experimenter showed a 5-minute video describing how to use the ITS and what tape diagrams represent to all

students. Next, students practiced equation solving using their randomly-assigned ITS version for approximately

15 minutes. On the second day, students started the class by solving equation problems in the assigned ITS for

approximately 15 minutes. After working with the ITS, students took the web-based posttest for 15 minutes.

Students were given access to both ITS versions a week after the study.

Results

Pre-post test results

One 6th grader was absent for the second day and excluded from the analysis; therefore, we analyzed data from

the remaining 108 students, namely, 54 6th-graders (28 Diagram, 26 No-Diagram) and 54 7th-graders (27 Diagram,

27 No-Diagram). Open-ended items were coded for whether student answers were correct or incorrect by two

researchers (Cohen’s kappa = .91). Table 2 presents raw pretest and posttest performance on conceptual

knowledge (CK) and procedural knowledge (PK) items. The maximum scores were 9 and 4, respectively.

Table 2: Means and standard deviations (in parentheses) for CK and PK on the pretest and posttest

CK (maximum score: 9)

PK (maximum score: 4)

Pretest

Posttest

Pretest

Posttest

Diagram

3.53 (1.56)

3.80 (1.94)

1.51 (1.17)

1.73 (1.39)

No-Diagram

3.42 (2.02)

4.01 (2.27)

1.55 (.92)

1.83 (1.57)

We first tested hypothesis H1 (benefits of anticipatory diagrammatic self-explanation with respect to

learning outcomes). We analyzed the data using hierarchical linear modeling (HLM) because the study was

conducted in nine classes taught by four teachers at two schools. According to both AIC and BIC, a two-level

model showed the best fit, in which students (level 1) were nested in classes (level 2). The inclusion of teachers

(level 3) and schools (level 4) did not improve the model fit. We ran two HLMs with posttest scores on CK and

PK as dependent variables, type of ITS assigned as the independent variable, and pretest scores (either CK or PK

given the dependent variable) as a covariate. For both CK and PK, there was no significant effect of the

Diagram/No-Diagram condition (CK: t(99.3) = -1.030, p = .31, PK: t(99.4) = -0.292, p = .77). We also ran two

additional HLMs, regressing pretest-posttest gains for CK and PK (dependent variables) on type of ITS. There

was a significant gain from pretest to posttest for CK (t(108) = 2.778, p < .01) but not for PK (t(106) = 1.153, p

= .26), and no significant effect of ITS type. This suggests that students in both ITS conditions improved in

conceptual knowledge but not in procedural knowledge.

We then analyzed the strategies that students used to solve the problem-solving items on the pretest and

posttest. We adopted a coding scheme by Koedinger et al. (2008), which identified both formal (algebraic) and

informal (non-algebraic) ways of solving equations (Cohen’s kappa = .73; Table 3). We were primarily interested

in the Algebra strategy because the goal of the ITS was to help students learn the formal algebraic strategy. We

performed the strategy coding independent of the correctness coding used to calculate students’ test scores. On

the pretest, 11 students in the Diagram condition and 17 students in the No-Diagram condition used the Algebraic

strategy on one or more problem-solving items. More students did so on the posttest; 26 students in the Diagram

condition and 23 students in the No-Diagram condition used the Algebraic strategy. We used McNemar’s test to

compare the frequency of use of the Algebra strategy at pretest and posttest for each condition. The increase in

frequency was significant (p < .01) for students in the Diagram condition but was not significant (p = .11) for

students in the No-Diagram condition. This pattern also held when we limited the analysis to problems involving

negative numbers (transfer problems); there was a pretest-posttest increase of only 1 student in the No-Diagram

condition, but 12 students in the Diagram condition (p < .01). These findings suggest that, although students who

learned with anticipatory diagrammatic self-explanation did not have greater gains on tests of conceptual and

procedural knowledge, they were more likely to learn the formal algebraic strategy and to apply it to problems

with no diagram support, even for problem types that they did not practice in the ITS (H1 partially supported).

Table 3: Strategies used to solve equations, adapted from Koedinger et al. (2008)

Strategy name

Description

Example answer for 3x + 2 = 8

Algebra

Student uses algebraic manipulations to find a solution

3x = 6

x = 6/3 = 2

Unwind

Student works backward using inverse operations to

find a solution

8 – 2 = 6

6/3 = 2

Guess and Check

Student tests potential solutions by substituting

different values

3*2 + 2 = 8

6 + 2 = 8

Other

Student uses other non-algebraic strategies

3 + 2 = 5

8/5 = 1.6

Answer Only

Student provides an answer without showing any

written work

x = 2

No Attempt

Student leaves problem blank or explicitly indicates

that she/he does not know how to solve the problem

“I don’t know”

Log data analysis on students’ learning processes

Next, we tested hypothesis H2 (benefits of anticipatory diagrammatic self-explanation with respect to learning

processes), using log data from the ITS. Specifically, we looked at “learning curves”, which plot students’

performance within the ITS over time (Rivers et al., 2016). Figure 5 depicts learning curves for the two conditions.

The y-axis shows the error rate on steps in tutor problems, averaged across students and skills, and the x-axis

shows the sequence of opportunities for practicing each skill. Learning curve analysis assumes that learning

occurs when a curve starts with a relatively high initial error rate and gradually goes down as students practice

the target skills. The curves are fit to student performance data using the Additive Factors Model (AFM), a

specialized form of logistic regression (Rivers et al., 2016). In our study, students practiced a variety of equation-

solving skills (e.g., subtracting variable terms). We expected that students who learned with diagrammatic self-

explanation support would perform better in the ITS than their peers who did not receive the support (H2). On the

symbolic problem-solving steps in the ITS (i.e., excluding the performance on the self-explanation steps, which

only occurred in the Diagram condition), students in the Diagram condition had a lower error rate than students

in the No-Diagram condition. Figure 5 shows learning curves averaged across all symbolic equation-solving skills

students in both conditions practiced. Students in the Diagram condition made fewer errors than those in the No-

Diagram condition, especially on the earlier opportunities. Both groups improved as they solved more problems

(i.e., both curves show a gradual decline). After much practice, the No Diagram condition eventually lowered

their error rate to the same level as the Diagram condition.

In parallel to the trend observed in the learning curves, we found that, when restricting the analysis to

symbolic steps only (i.e., excluding diagrammatic self-explanation steps), students who received the self-

explanation support trended toward using fewer hints (t(89.52) = -1.812, p = .07) and spent significantly less time

on each symbolic problem-solving step (t(99.51) = -2.238, p = .03) than students who did not receive the self-

explanation support (Table 4). The average number of problems solved in the ITS during the (fixed amount of)

available time did not differ significantly across conditions, (t(99.30) = -0.528, p = .60) (Table 4).

In summary, the students in both conditions practiced a similar number of problems in the ITS in a similar

amount of time overall, and the anticipatory diagrammatic self-explanation helped students spend less time and

ask for fewer hints on symbolic steps (H2 partially supported). In addition, the learning curves indicate that

students in both conditions learned equation-solving skills eventually, but the students in the Diagram condition

learned them faster and had a smoother experience, with fewer errors.

Table 4: Average number of problems solved, number of incorrect attempts, number of hint requests, and average

time spent on symbolic steps in the ITS (standard deviation).

Average number of

problems solved

Average number of hints

requested per step

Average time spent per

step

Diagram

15.40 (9.02)

0.68 (0.96)

15.99 (9.91)

No-Diagram

16.17 (11.16)

1.02 (1.38)

20.27 (13.95)

Figure 5. Learning curves for the Diagram condition (red) and the No-Diagram condition (green) averaged

across the skills students practiced during the symbolic problem-solving steps. Dark and light blue lines show

predicted curves based on the AFM (dark blue: Diagram condition, light blue: No-Diagram condition).

Discussion and Conclusion

Self-explanation has been shown to support student learning in various domains, but it is not easy to design

appropriately-scaffolded self-explanation activities. Our study investigated the effectiveness of anticipatory

diagrammatic self-explanation as a proposed approach to enhancing both learning and performance. We found

that anticipatory diagrammatic self-explanation embedded in an Intelligent Tutoring System (ITS) helped students

learn to apply a formal, algebraic problem-solving strategy to problems outside the ITS and to transfer problems

involving negative numbers (H1). Anticipatory diagrammatic self-explanation also supported student

performance within the ITS, measured by lower learning curves, less frequent use of hints, and less time spent on

each symbolic equation-solving step (H2). Anticipatory self-explanation did not lead to differences in posttest

scores, contrary to H1, but it helped students learn more efficiently; students learned the formal algebraic strategy

while solving a similar number of problems with less time and fewer errors and hint requests, and they achieved

similar gains on conceptual knowledge (H2).

We attribute these findings to the design and learning principles used in supporting anticipatory

diagrammatic self-explanation. Specifically, we reason that the process of selecting the next correct-and-strategic

problem-solving step, depicted diagrammatically, helped students perform better and faster on the corresponding

step with symbols. On steps with symbols, students had a diagrammatic representation of the step available to

them on the screen. They could refer to this representation as they sought to express the step using mathematical

symbols. Engaging in this cognitive process may have helped students understand step-level formal strategies in

a visual form (e.g., visually seeing that constant terms are taken out from both sides of an equation). Comparing

and contrasting the different tape diagrams may have supported students in selecting steps that were both correct

and strategic, and it may have helped them avoid using informal strategies, such as guessing. It may be, as well,

that the better performance resulting from the anticipatory diagrams gave students a bit more confidence to take

on the challenge of moving towards formal algebra.

An intriguing question is why the ITS with anticipatory diagrammatic self-explanation did not lead to

greater gains in conceptual and procedural knowledge than the ITS with no diagram support. Regarding

procedural knowledge, students did not make gains from pretest to posttest in either condition. Further, there was

no difference in solving equations correctly between the conditions at post-test, even though students with

diagrams exhibited greater use of formal problem-solving strategies. It is possible that students in the Diagram

condition might need more practice in correctly applying the formal strategy they acquired in the ITS without the

help of diagrams. In other words, it seems that students in the Diagram condition developed further towards formal

use of algebra than their counterparts in the No-Diagram condition, but not yet to the extent that the use of the

more challenging formal strategies paid off in terms of improved correctness. Regarding conceptual knowledge,

it might be that the anticipatory use of diagrams in the ITS focused students primarily on strategic issues, as the

diagrams were used in planning problem-solving steps. It may be that students need to engage in “principle-based

explanation” (Renkl, 1997) to facilitate acquisition of conceptual knowledge (e.g., verbally explaining why the

selected diagram is correct and strategic). It might also be that students with varying levels of prior knowledge

benefit from diagrammatic self-explanation differently (Booth & Koedinger, 2012). Future studies should

examine the effects of anticipatory diagrammatic self-explanation with students having varying degrees of prior

knowledge and experience in algebra.

Our study has several limitations. First, the study was conducted with one specific type of diagrams, tape

diagrams, and it focused on one specific task domain, equation solving in algebra. To understand how the results

could generalize across domains and types of visual representations, more research is needed to examine the

effects of anticipatory diagrammatic self-explanations. Also, it is possible that students were not very motivated

to work on the posttest, especially given that the study was conducted remotely during the COVID-19 pandemic.

This may have contributed to the absence of pretest-posttest gains in procedural knowledge, even though the

learning curves suggest that learning occurred.

In summary, we designed a novel self-explanation scaffolding support for students in middle-school

algebra, namely, anticipatory diagrammatic self-explanation. We investigated the effectiveness of this support

embedded in an Intelligent Tutoring System, in a classroom study. We found that anticipatory diagrammatic self-

explanation helped students learn formal algebraic strategies and perform better on problem solving, while making

similar conceptual gains as students who did not receive the support. Our study contributes to the theoretical and

practical understanding of how visual representations, contrasting cases, and anticipatory self-explanation can be

integrated into scaffolding support that helps students learn and perform effectively and efficiently.

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Acknowledgments

This research was supported by NSF Award #1760922 and by the Institute of Education Sciences, U.S.

Department of Education, through Award #R305B150003 to the University of Wisconsin–Madison. The opinions

expressed are those of the authors and do not represent views of NSF or the U.S. Department of Education. We

thank Max Benson, Susan Brunner, Octav Popescu, Jonathan Sewall, and all the participating students and

teachers. Copyright 2021 International Society of the Learning Sciences. This is an author’s copy.