Analogical Scaffolding and the Learning of Abstract Ideas in Physics : Empirical Studies

This paper reports on empirical studies of students’ use of analogy in learning physics, focusing on the role of representations. To study the utility of the analogical scaffolding model, it is applied to design curricular materials using multiple analogies, and successfully employed to predict the outcomes of two studies based on these materials. Students in three treatment groups were taught EM waves concepts using multiple analogies, waves-on-a-string and then sound waves. Different representations were used in the materials for each treatment group. One group used abstract representations, one used concrete representations, and a third used both abstract and concrete (i.e., blends). In both studies, students presented with materials using blended representations (those consistent with the analogical scaffolding model) outperformed students using abstract representations. In the first study which examines multiple analogies, students in the blend group outperform the students in the abstract group by as much as a factor of three (73% vs. 24% correct, p=0.002). In the second study, examining representation use within one domain (sound waves), the blend group outperforms the abstract group by as much as a factor of two (48% vs. 23% correct, p=0.002). Data also confirm the utility of the model to explain when and why students succeed and fail to use analogies and interpret representations appropriately. PACS: 0140FK © PER@C Group, University of Colorado, 2007


Introduction
This paper examines the interplay between two essential components of scientific reasoning, representation and analogy.Scientists use multiple representations (including verbal, graphical, and gestural) and easily shift among these representations.[1] [2] Scientists also frequently generate and use analogies to reason and communicate in day-to-day activities.[3] Representation and analogy are often considered convenient ways of communicating concepts, but with the implication that concepts transcend these forms of discourse.This view is controversial.[4] Previously, we have proposed a model of student reasoning which combines the roles of representation, analogy, and layering of meaning -analogical scaffolding.[5] The present empirical studies build on this model to examine its utility.In this paper we present a series of results demonstrating the vital intertwining of representation, analogy, and conceptual learning in physics.
Analogy plays an essential role in scientific reasoning.[6] Historical examples include Rutherford's planetary model of the atom [7] or Maxwell's application of fluid theory to electromagnetism.[8] Analogies are commonly used to teach students physics, as evidenced by the range of analogies used in physics textbooks.[9] Research investigating students' use of analogy in physics gained momentum during the early 1980's.Early models proposed that an analogy can be treated as a mathematical mapping from a familiar conceptual structure, the base, to an unfamiliar conceptual structure, the target.[10] [11] Using this framing of analogy, Gentner and Gentner [12] found that students' reasoning about electric circuits was measurably influenced by the analogies that these students generated (i.e., flowing water vs. moving object analogies), demonstrating that analogies constitute more than mere surface terminology; indeed, analogies generate inferences.However, this study also found that analogies that were taught to students were not as influential on these students' reasoning compared to analogies that the students generated themselves.Some question the notion of mapping between well defined structures, [13] [14] suggesting this framing of analogy is insufficient to explain the complex (and often fragmented [15] [16]) ways students reason.Nonetheless, researchers have confirmed that analogies can be productive for student learning, documenting cases where analogies generated by students [17] and expert physicists [18] can contribute to productive reasoning about physics problems.The idea that analogies can play a significant role in student reasoning is now well supported.Spiro et al [19] suggest teaching with multiple analogies in order to circumvent the drawbacks of single analogies, (e.g., single analogies may be misleading or incomplete) especially when teaching complex and difficult topics.However, finding consistently productive ways of teaching with analogies remains a challenge to researchers.
Teaching with analogies has met with mixed success [12] [14] despite efforts to directly teach step-by-step processes, [20] [21] or to foster student use of analogies.[22] [23] Recently, however, some progress has been made toward identifying possible mechanisms of student analogy use.Several lines of research have suggested a tradeoff between within-domain and across-domain learning of abstract principles (e.g., modulo-3 arithmetic [24]).This tradeoff appears to be coupled to the concreteness of the representations used to teach students.[24] [25] Here, the concreteness of a representation is gauged by the degree to which the representation contains salient, information-rich features (e.g., a picture of a soccer ball is considered more concrete than a black dot meant to represent a generic rolling object).While researchers find that concrete representations are more productive for students learning within a single domain, the use of abstract representations better facilitates students productively using those ideas in a second domain.Along these lines, Van Heuvelen and Zou [26] successfully used concrete representations to scaffold students interpretations of abstract (i.e., mathematical) representations when solving work-energy problems.Interestingly, Goldstone and Sakamoto [25] report this tradeoff in learning for "low-achieving" students, but they find little or no such effect for "high-achieving" students.Sloutsky et al [24] find that irrelevant concreteness (e.g., pictures of insects used to represent mathematical entities) can hinder across-domain learning of mathematical principles.Surprisingly, Goldstone and Sakamoto [25] find that even relevant concreteness can hinder across-domain learning.
These recent results parallel our own findings that representations can play a key role in teaching students with analogies.[27] Based on these findings, we proposed a model of analogy use, analogical scaffolding, [5] which describes mechanisms by which multiple analogies may be layered in order to learn abstract ideas.According to this model, concrete and abstract representations play key, complementary roles in this layering process.Based on this model, we have modified curricular materials aimed to teach college physics students about electromagnetic (EM) waves by using analogies.Ambrose et al [28] have identified a number of student conceptual difficulties with EM waves in order to develop curricular interventions.These prior findings call for further study of how students can learn this challenging topic, particularly with regard to the use of wave representations.The present paper describes two studies to investigate the effectiveness of the analogical scaffolding model to teach students about EM waves, focusing on the role of representations in promoting (or impeding) the productive use of analogies by students.
In preliminary work [5] we found that students taught EM waves concepts using materials based on analogical scaffolding outperformed students taught the same EM waves concepts without analogies on a pre-post assessment.In this paper, we describe two follow-up studies, the first examining student learning across multiple conceptual domains and the second examining student reasoning within a single domain.Primarily, we consider analogical scaffolding to be a cognitive model and these studies seek to examine the utility of this model to explain student reasoning and responses in educational environments.
In the across-domain study, we taught students about EM waves using analogies from multiple domains (wave-on-a-string and sound waves).We explore the implications of varying the concreteness of the representations used to teach students in an algebra-based introductory physics course.In this study we ask, how does the model explain (and predict) student learning under different conditions, i.e., using different representational forms to teach?As a secondary goal, we may investigate analogical scaffolding as a teaching intervention.To this end, we ask whether students taught with materials designed according to an analogical scaffolding framework demonstrate significant learning gains.[29] In the within-domain study, we explore the implications for student reasoning and use of analogy by varying the concreteness of the representations used on a quiz in a single conceptual domain, sound waves.We administered these quizzes to students in another algebra-based introductory physics course in order to examine a within-domain application of analogical scaffolding.We pose the following research questions for this second study.(1) Previously, we found students associated different representations of a sound wave with various conceptions of sound waves (e.g., though sound is a longitudinal wave, students associate a sine wave representation of sound with transverse wave motion).[27] In the present within-domain study we explore the directionality of this association and ask: do representations drive student use of analogy and, by proxy, conceptions of sound waves?(2) How does varying the concreteness of representations affect students' reasoning about sound waves?The findings of this second study give insight into student learning of sound waves, one of three analogical scaffolds we pose as productive for layered student learning of EM waves in the preliminary [5] and across-domain studies.
In both the across-and within-domain studies, we find students demonstrate markedly different performance depending on the form of representations used to teach.In the acrossdomain study, the model makes accurate predictions about student performance and, importantly, predicts which form of analogical scaffolding (of three investigated) is optimal for student learning of EM waves.Students taught with a curriculum aligned with appropriate analogical scaffolding demonstrate significant learning gains in a particularly challenging content area, EM waves.In the within-domain study, we find representation can drive student reasoning about analogies.Furthermore, the analogical scaffolding model predicts which representational forms (and combinations of these) are optimal for students to make productive use of an abstract representation of a sound wave, such as a sine wave.

Analogical Scaffolding
We briefly outline analogical scaffolding theory -a more detailed account is presented in a prior paper.[5] The analogical scaffolding model draws on theories of representation, [30] conceptual blending, [31] and layering of ideas, [32].We draw on the work of Roth and Bowen [30] to describe the relationship between a signifier, sign, the thing the sign refers to, referent, and a knowledge structure mediating the sign-referent relationship, schema.The word sign refers to external representations, such as text, graphs, equations, pictures, gestures, or utterances.
Schemata (plural of schema) can be considered knowledge structures employed to interpret signreferent relationships.Each system of sign-schema-referent can be considered a mental space, defined by Fauconnier and Turner as "a small conceptual packet constructed as we think and talk, for purposes of local understanding and action."[31, page 102] According to this model, productive schema elements are coupled to a sign whose surface-level features are associated with these schema elements.(Along the lines of what-you-see-is-what-you-get, or WYSIWYG, as described by Elby [33].)For example, consider a picture of compressed and rarefied air particles, a sign representing the referent sound wave (Figure 1a).The sign and referent are coupled to a schema containing the elements "longitudinal" and "disturbance spreading through space" or "3D".Now, consider a sound wave represented by a sine wave (Figure 1b).The surface-level features of a sine wave are more tightly coupled to schema elements such as "transverse" and "2D" (a sine wave is generally drawn in a single plane).[27] In this case, the sine wave can cue a schema that is unproductive for sound.However, these two sign-referentschema systems may be blended, whereby the sine wave comes to be coupled to a schema containing "longitudinal" and "3D".This process constitutes one layer within a conceptual domain.If, in a subsequent layer, the sine wave is coupled to the referent EM wave, the 3D longitudinal schema may be inherited by the EM wave mental space via another blend.Blends combine mental spaces, linked by some connection between these spaces (e.g., the same sign or sometimes the same referent with different signs), and project selected schema elements (e.g., "3D") from these mental spaces to generate a blended space.Increasingly complex and abstract ideas can be built up by a series of blended layers.[32] For instance, a wave-on-a-string blends with sound waves, building up to EM waves.As a concrete example, in the next section we will describe a detailed application of this model to predict the outcomes of two empirical studies of student learning of E/M waves by layered analogies of string and sound waves.

Classroom Setup
The participants in the across-domain study were 152 college students enrolled in the second-semester of an algebra-based introductory physics course, focusing largely on electromagnetism.The first semester of this course included instruction on waves-on-a-string and sound waves as well as general wave properties.Prior to the tutorial activities described here, lectures and homework had covered electric and magnetic fields, but had not yet covered EM waves.This typical introductory course consists of three 50-minute lectures, used the Touger text [34], an online HW system [35], and included one 2-hour recitation each week.Recitations generally included laboratory activities, but on occasion students worked on pencil and paper tutorials in lieu of hands-on experiments.Students generally worked in groups of three to five.
During these tutorial activities, the teaching assistant roamed the classroom answering students' questions and probing students' understanding of the materials to be learned.In the across-domain study, groups of students within a given recitation section were assigned to one of three treatment groups, denoted abstract, concrete, and blend.Table 1 lists the number of students (N) for each group, summed over all recitation sections, and the average course grade for students in each group.We found no statistically significant difference between the average grades for the three groups (p>0.3,2-tailed z-test [36]).In recitation, students were issued pre-and post-tests on EM waves immediately before and after tutorials.These assessments were identical in all three treatment groups.The pre-test was administered at the beginning of recitation.These were collected, and students were then divided into three groups, each receiving a different version of a tutorial on EM waves, described below.After completing the tutorial, students were issued a post-test, identical to the pre-test.
The only difference in treatment between the three groups was the type of representation used in the tutorial.(The assessments used identical representations for all three groups.) The assessments consisted of two open response questions, shown in Figure 2.These questions are based on the materials used by Ambrose et al [28] to evaluate the Tutorial [37] on EM waves, but were modified based on student interviews [38] Since these questions were open response, there was a large range of possible answers, and the likelihood of students guessing the correct answer was extremely low.Students were asked to explain their reasoning on each question.Note that these two questions require that students interpret the pictures in Figure 2 as representing a snapshot in time of EM plane wave traveling to the right.Thus, the correct answer to question 1 is I=J=K=L, since for a plane wave the magnitude of the electric field depends only on the x-coordinate.The correct answer to question 2 is P=Q=R=S, since, for a plane wave, the magnitude of the electric field depends on x, but since this is also a traveling wave, the time average signal is independent of x.Note that if question 2 had asked for the magnitude of the electric field at this instant in time, the correct answer would be P=Q=R>S, with S having magnitude zero.

Tutorials
The tutorials used in the across-domain study were based in part on The Tutorials in Introductory Physics, [37] but were modified to teach about EM waves using wave-on-a-string and sound wave analogies.[39] The tutorials consisted of three main parts.Part 1 used a wave on a string to introduce transverse and traveling wave ideas.Part 2 used sound waves to introduce The tutorials in each experimental group were nearly identical in content and wording, but differed in the representations used.Figure 3 shows a subset of the representations of string, sound, and EM waves used in the abstract, concrete, and blend groups.The complete set of tutorials and surveys can be found in supplementary materials.[39] The canonical wave representation is a sine wave, which we consider an abstract representation [5].In the abstract group, a sine wave is used consistently to represent string, sound, and EM waves.
[40] The representations used in the concrete group include more salient features, for instance, showing compressed and rarefied air particles in a sound wave spread throughout space.In the blend group, students were presented with both abstract and concrete representations simultaneously.
The tutorials included significant framing for students to make sense of these representations and learn about EM waves, generally in the form of Socratic questioning written into the tutorials.
This framing (and wording) was nearly identical for the three treatment groups.

Predictions
We claim the analogical scaffolding model makes correct predictions of student answers (and reasoning) to explain the results of the across-and within-domain studies.Before outlining these predictions of analogical scaffolding, we explore alternative models.According to one model, students' prior knowledge consists of relatively stable and well formed structures, akin to scientific theories, [41] [42] [43] that are not strongly linked to particular contexts.[44] When these ideas are non-canonical they are called misconceptions.[33] This model predicts that many students will apply these theory-like ideas to conceptual questions about EM waves, often resulting in students answering these questions incorrectly.The way misconceptions are changed is that students are presented with cases that conflict with their prior knowledge, and these students therefore reorganize their knowledge to align with this new case.This model has merit, and in fact the tutorials used in the across-domain study do address common student ideas about EM waves which may be inconsistent with experts' ideas.However, suppose students are presented with cases that conflict their prior knowledge in a tutorial, but under different representational conditions, as is the case in the present study.This model of student learning alone does not predict nor does it explain how students will reorganize their knowledge under these different conditions.In other words, this model is not sensitive to context -it predicts that students will reorganize their knowledge in all three treatment groups (abstract, concrete, and blend), but does not make specific predictions about how students will learn differently in the three groups.Determining which condition is optimal is purely an empirical endeavor.To be sure, curricular materials based on this model can be extremely productive in bringing students ideas closer to experts'.However, we seek mechanisms which are sensitive to context and can therefore predict and explain how using different representations and analogies impact student learning.
Elby [33] proposes one such mechanism of interpreting representations, what-you-see-iswhat-you-get or WYSIWYG.When WYSIWYG is activated, students interpret representations literally, for instance, they may interpret a graph shaped like a hill literally as a hill (even if this graph is of velocity vs. time).We predict that on questions like those in Figure 2, students who apply WYSIWYG will treat the sine wave literally as moving up and down in the x-y plane.This prediction is based most significantly on the particular representations used on the pre-and posttests.However, because WYSIWYG is so strongly tied to these representations, it may fail to predict when students will not use WYSIWYG.WYSIWYG alone would predict no differences between the three conditions on the pre-test, but also on the post-test, since all three groups had exactly the same questions with the same representations.One explanation for why students would not use WYSIWYG is simply that student answers have some randomness to them, or alternatively that students who answer with a non-WYSIWYG interpretation (but correctly) simply "get it".Analogical scaffolding uses WYSIWYG as one mechanism of learning from representations, but additional mechanisms are required to explain when and why students will not use WYSIWYG, especially when students use other interpretive strategies productively.In other words, analogical scaffolding explains why some students appear to "get it", but also why students who do not answer concept questions correctly may nonetheless answer these questions in predictable ways.
Studies of analogy suggest that while potentially powerful, students often fail to use an analogy productively if at all.Therefore, we might expect students to directly apply what they have learned about EM waves during the tutorials to the post-test question, but not use ideas from string and sound waves.That is, students do not apply the analogies provided.In this case, we might expect differences between the abstract, concrete, and blend groups based on the treatment of EM waves in the tutorials.However, WYSIWYG applied to EM wave representations (both sine wave and vectors) does not lead in any obvious way to developing 3D or traveling wave ideas about EM waves, since these ideas were taught only for sound and wave on a string.Again, if students only directly applied their (possibly reorganized) knowledge of EM waves to the post-test question, we would not expect differences between the three groups since these questions specifically test students knowledge of 3D and traveling waves.
These alternative models can be reformulated according to three hypotheses on the role of representations in student learning EM waves via anaological scaffolding: • The null hypothesis: Student learning depends mostly on prior knowledge and reorganizing this knowledge to align with a new conflicting case.Representations and student learning are largely independent, both within-and across-domains, and we should therefore expect no differences between the three groups in both the across-and withindomain studies since the only variation between conditions was the representations used.
• The weak hypothesis: representations do couple to students' prior knowledge along the lines of WYSIWYG, but this coupling is only dependent on the immediate context.
Observed differences between the treatment groups in the within-domain study would be sufficient to confirm this hypothesis, since students in this study received different representations on the assessment.However, this hypothesis would also predict no measurable differences between treatment groups in the across-domain study, since all students received the same representations on the assessments.
• The strong hypothesis: representations not only cue existing prior knowledge, but also lead to the dynamic formation of new knowledge.This process is strongly dependent on the form and presentation of the representations.To confirm this, we would need to observe differential performance between the treatment groups in both the within-and the across-domain study.Differential performance would show that the representations used to teach had different effects on how students learn new interpretations of the representations in Figure 2.
We now apply analogical scaffolding theory to predict the outcomes of the across-domain study.Three possible sequences of representational cueing, blending, and projection are shown in Figures 4 and 5.The sign (representation) is shown at the upper right node of each triangle, referent at the upper left node, and schema at the bottom node.Figure 4 represents the abstract tutorial, with only sine wave representations used, and the concrete tutorial, with concrete representations used.In the abstract tutorial, the surface level interpretations of the sine wave would lead students to use a schema including the features 2D and "up means up".[27] This schema is projected through to EM waves, cued in each layer by the same sine wave.
Alternatively, in the concrete tutorial, surface level interpretations lead students to apply different schemata to string, sound, and EM waves.However, students are predicted not to project these schematic elements (e.g., 3D) from one domain to the next as often as in the blend group since there is no corresponding sign (e.g., sine wave) to cue blended schemata in subsequent layers.This predicted lack of projection, one possible approach students may take, is represented by dashed arrows in Figure 4.
Note that in Figure 4, schemata are presented as separate, unblended pieces.In these cases, WYSIWYG operates within each piece, cueing schemata that are tightly coupled to signs.
In the blend tutorial (Figure 5), these schemata are cued by signs in a similar fashion to the abstract and concrete tutorials, but in this case schemata blend.Blended schemata then project through each layer and subsequently reblend.Each blend corresponds to an additional node between the sign (upper right) and schema (bottom) nodes of the resulting triangles.The final blend for EM waves has three nodes, corresponding to three prior blends.Note that in the blend treatment group, the schemata resulting from each blend are non-WYSIWYG.
This model predicts that students in the abstract group will be most likely of the groups to apply 2D, "up means up" object-like schema elements when answering the post-test questions.
This reasoning would be consistent with the answer I>J>K>L on question 1. (This reasoning would also be consistent with a number of incorrect answers to question 2 based on "up means up" reasoning, for instance P>Q=S>R, P>Q>S>R, P>Q=S=R, etc.).Students in the blend group will more likely apply 3D, time-varying schema elements, and treat the sine wave as representing an abstract quantity (e.g., field) rather than treating the sine wave as an object (i.e., an object that goes up and down in space like a string).This reasoning would be consistent with I=J=K=L on question 1 and P=Q=R=S (both traveling and 3D) or P=Q=R>S (only 3D) on question 2.
Students in the concrete group will fall somewhere in between, having been exposed to the essential schema elements, but not led to create blends of these schemata.Pre-test results for all groups would be most similar to the post-test predictions for the abstract group, since students are asked to answer questions about a sine wave representation of an EM wave before instruction.Note that we do not expect these coarse categorizations to describe individual students, as individual student resources and reasoning are sure to vary.We therefore note that these predictions are probabilistic and we predict trends in students reasoning for statistically robust numbers of students.(Our studies use N>100 subjects.)

Question 1
Key results from the pre-and post-tests in the across-domain study are shown in Figures 6 and 7.
On pre-test question 1 (Figure 6), less than 10% of students answered correctly (I=J=K=L), with no statistically significant difference between groups (p>0.7).On the post-test, the scores for all groups increased, but students in the concrete and blend groups outperformed students in the abstract group by factors of more than two and three, respectively (p<0.01).Further, students in the blend group outperformed students in the concrete group by a margin of 16% (p<0.1).
The most popular (incorrect) answer on pre-test question 1 was I>J>K>L, answered by approximately 24% of students.This pre-test result was the same, statistically, in all three groups (p>0.3).On the post-test, less than 10% of student in the concrete and blend groups answered I>J>K>L, while 18% of students in the abstract group wrote this answer.This result is statistically significantly different between abstract and blend groups (p=0.055),but not between abstract and concrete groups (p=0.22).Another somewhat popular answer, J>I=K>L, was produced by 18% of students on the pre-test.On the post-test, 22% of students in the abstract group answered J>I=K>L, while less than 5% of students in the concrete and blend groups wrote this answer, significantly less than the abstract group (p<0.01).We note that these incorrect answers are similar to those observed by Ambrose et al. [28] We coded student explanations of reasoning on question 1 according to 5 categories, shown in Table 2.The number of students (N) producing each answer is shown above the corresponding answer, with the percentage of students producing that answer binned into each reasoning category below.We used an emergent coding scheme based on students' answers.
Proximity to Line corresponds to primitive reasoning [45] such as "closer is more", i.e., interpreting the sine wave as an object and reasoning that a closer proximity to this object means stronger field.Read as Graph corresponds to primitive reasoning such as "higher is higher", i.e., interpreting the positions of the points as heights on a graph of amplitude.[46] Same X Position corresponds to reasoning that the magnitude of the E-field depends only on the x-coordinate.
Sound Words indicates usage of words related to sound, such as pressure and density.Other corresponds to explanations that were rare, unintelligible, or left blank.We group all students together on the pre-test, since these responses came before any differential instruction.to more than 33% on the post-test.Importantly, students who answered question 1 correctly used similar reasoning across all three groups.We note that very few students in the concrete and blend groups answered I>J>K>L or J>I=K>L, but that for these few students, their reasoning patterns match those of students in the abstract group.However, combining these results with Figure 6, we find that students in the concrete and blend groups were significantly more likely to answer correctly and, thus, more likely to use Same X Position reasoning on the post-test compared to students in the abstract group.

Question 2
Figure 7 shows the results for question 2. On the pre-test, less than 10% of students answered correctly (P=Q=R=S), with no statistically significant difference between groups (p>0.6).This question proved challenging for students, and less than 18% answered correctly on the post-test (no difference between groups, p>0.3).We did, however, find significant results on another popular answer, P=Q=R>S, which would be correct if the question had asked for the magnitude of the E-field at the instant shown.We consider this answer partially correct, since it includes the plane wave feature of EM waves, but not the traveling wave feature.On the pre-test, students in the abstract group produced the partially correct answer more often than the other two groups (p<0.06).This trend reversed on the post-test -the blend group produced the partially correct answer significantly more often than both the concrete group and abstract group (p<0.02).Interestingly, the fraction of students in the abstract group answering partially correct was unchanged from pre-to post-test, but the majority of these students answering partially correct on the post-test were not the same students that selected the partially correct answer on the pre-test.[48] Further Analysis & Follow-up studies We found that students in the blend group made the greatest shifts to the correct (or partially correct) answers, followed by the concrete group, with the abstract group performing the lowest.As a direct measure of changes in student reasoning, we found that 40% of students in the abstract group did not change their answers to question 1 from pre to post, while less than 20% of students in the other two groups did not change their answers from pre to post.On question 2, 15% of the abstract group did not change their answers from pre to post, while less than 8% of students in the other groups answered the same way from pre to post.
Two weeks after the EM waves tutorial, students were issued an online quiz with one question directly targeting the EM wave concepts in the tutorial.Lectures and homework during this two week interval included material covering EM waves.On this follow-up question, we found the same trends as we did on the post-test.Students from the blend group outperformed students from the concrete group (p=0.2), and both of these groups outperformed the abstract group (p<0.05).Thus, students in the blend group not only outperformed the other groups immediately following the tutorial, but these trends in differential performance persisted over the long term.Evidently, the instruction during this two week gap, the same for all students, did not help students taught only with abstract representations catch up with students who were taught with concrete representations (or both abstract and concrete in the blend group).
Goldstone and Sakamoto [25] found that varying the concreteness of representations affected the learning of low-performing students, but that high-performing students were relatively unaffected by this variation.We explored the possibility of finding a similar result by analyzing the preceding results for the upper and lower halves of the class (high-and lowperformers, respectively) based on overall course grade.Overall, we found no significant differences between the pre-post results of high-and low-performers across all three treatment groups, with one exception.High-performers in the concrete group performed the same as students in the blend group, but low-performers in the concrete group performed less well than the blend group (but still better than the abstract group).Thus, we find that teaching with multiple representations (blend group) may benefit low-performing students compared to teaching with single concrete representations.At the same time, teaching with single abstract representations appears to limit learning even for typically high-performing students in this population studied.

Discussion of the Across-Domain Study
In the across-domain study, three versions of a tutorial on EM waves used varying representations according to a model of analogical scaffolding, and the model was employed to predict trends in student learning with these tutorials.We demonstrated applications of the model to construct schematic representations of these tutorials (Figures 4 and 5) and to make specific predictions about student performance under these three conditions.The across-domain study demonstrated two key findings on student learning with analogies.( 1) Across several conceptual questions on EM waves, students taught with blend representations consistently outperformed students taught the same ideas using concrete or abstract representations only.Students taught with only abstract representations faired worse (sometimes dramatically worse) than other students.(2) We found that students' reasoning about and answers to these questions were associated in similar ways for the abstract, concrete, and blend groups.However, students who were provided blend representations demonstrated the highest performance on these concept questions.These results bolster the power of the analogical scaffolding model to predict differences in across-domain student learning under different conditions.
While we predicted the outcome that students taught with blend representations would demonstrate the highest performance, analogical scaffolding may also be employed to analyze cases where treatments were less productive for student learning of EM waves.Abstract representations may not provide students with the useful schemata for string and sound waves to apply to EM waves.Rather, we found students used surface-level reasoning to interpret the meanings of these representations (see Table 2), leading these students to apply unproductive schemata to EM waves.For instance, a surface-level interpretation of a sine wave leads students to read string, sound, and EM wave diagrams as "higher means higher" or "closer means stronger".Concrete representations do provide productive string and sound wave schemata for students, and we observe students applying these schemata to EM waves sometimes.However, without an abstract representation to blend, these schemata are less likely to be applied to EM waves compared to when students are presented with both concrete and abstract (i.e., blend) representations together.
In summary, the across-domain study examined broad scale applications of analogical scaffolding and demonstrated the model's predictive power for student learning across multiple domains.The within-domain study, described below, examines implications of the model within the domain of sound waves, and analyzes student reasoning within this single domain.

Classroom Setup
The participants in the within-domain study were 353 college students enrolled in the first-semester of an algebra-based introductory physics course, focusing on Newtonian mechanics.This is the same course sequence as in the across-domain study and has a similar structure to the second-semester course described above.Since both studies took place during the same semester, the two studies involved different students.Students were again assigned to one of three treatment groups, denoted as abstract, concrete, and blend groups.All students in a given recitation were assigned to the same group and issued a quiz on sound waves.In recitation the week prior to this study, students had completed a laboratory activity on sound.This activity involved using a microphone to take measurements of sound waves inside a long tube.[49] Lectures prior to this study had covered mechanical waves, but students had received no explicit instruction on plane (3D) waves.Differences among teaching assistants (TA's) were mitigated by distributing the treatment group assignments evenly among the TA's.Table 3 lists the number of students (N) for each group, compiled over all recitation sections, and the average course grade for students in each group.The average grade for the concrete group was not statistically different from the other two groups (p>0.27).The blend group's grades were higher than the abstract group, with weak significance (p=0.064).While this last difference in grades is weakly significant, this difference does not account for the variance we find in our results, and all following significant results remain so when normalized to account for this small variation in student grade.

Sound Waves Quiz
Students were issued a quiz on sound waves at the beginning of recitation.[39] The quizzes for the abstract, concrete, and blend groups were nearly identical in content and wording, but differed in the representations used.The quiz contained three multiple-choice questions.
Question 1 presented an abstract, concrete, or blend representation of a sound wave, corresponding to each treatment group, directly to the right of the question statement as shown in Figure 8.The text of question 1 was the same for all treatment groups.The analogy choices in question 1 draw on students' conceptions of a sound wave described by Hrepic.[50] Note the representations in Figure 8 are the same as those shown in the middle of Figure 3 for the sound part of the tutorial in the across-domain study.
Questions 2 and 3 are shown in Figures 9 and 10, respectively.For the abstract and blend groups, the representations used on question 2 were the same (a sine wave).For the concrete group, question 2 used a picture showing air particles.The wording of question 2 was the same for all three treatment groups.Question 3 was identical for all three groups in both wording and representation used.Questions 1 and 2 both appeared on the first page of the quiz, and question 3 appeared on a separate second page.The analysis is similar.Note that the only differential conditions for students occurred during the assessment.If we assume students' ideas are relatively stable and theory-like, a misconceptions model would predict no differences between the groups.WYSIWYG would predict differences between the groups on question 1 -students in the abstract group will choose transverse wave analogies, while students in the concrete group will choose longitudinal wave analogies.
However, in the blend group, students are presented with two representations, and WYSIWYG does not provide a mechanism for why students would apply WYSIWYG to one representation over another.Therefore, using WYSIWYG alone, we would predict an even distribution of transverse and longitudinal wave analogies in the blend group.On question 2, WYSIWYG predicts that the concrete group will be likely to answer correctly, since this information can be read directly from the diagram, but does not distinguish between the abstract and blend groups, both of which had the same representation on question 2. WYSIWYG predicts that all three groups will answer similarly (or with similar distributions of answers) on question 3, which was identical in all three groups.We apply analogical scaffolding to predict the outcomes of the within-domain study.On question 1, surface-level interpretations of signs couple to associated analogies.Students in the abstract group, presented with a sine wave, will preferentially select analogies that involve vertical motion (e.g., crowd and string analogies) while students in the concrete and blend groups, presented with a picture of air particles, will preferentially select analogies that involve horizontal motion (e.g., slinky and football).[51] Students' surface-level interpretations of these signs will play key roles for questions 2 and 3. Students in the abstract group will use "up means up" reasoning, likely answering 1>2=4>3 on question 2. Students in the concrete and blend groups will also use surface-level reasoning, but in this case students will interpret the sign (air particles) as meaning the pressure is the same where the air particle density is the same (and therefore are likely to answer correctly, 1=2=3>4, or possibly take a more literal reading of the picture and answer 1=2>4=3).Importantly, students in the concrete group can map this information directly from the picture on question 2, while students in the blend group must interpret the sine wave in question 2 as standing for a 3D sound wave.On question 3, absent an overt sign, students' choice of analogy on question 1 will play a key role.Students in the abstract group will answer vertical motion ("up and down") while students in the concrete and blend groups will answer horizontal motion ("to the right" or "left and right").

Question 1
Figure 11 shows the six most popular single (or combination of) analogies selected by students according to their assigned treatment group, accounting for more than 92% of student responses.Overall, the slinky analogy was the most popular choice, accounting for 43% of all student answers.[52] If we include students who selected the slinky analogy in combination with others, we find 66% of students selected the slinky analogy.We find significant differences, however, between treatment groups in Figure 11.Students in the concrete and blend groups were significantly more likely to select the slinky analogy than students in the abstract group (p<0.01).
The concrete group was significantly more likely than the abstract group to select both football and slinky in combination (p=0.03), and the abstract group was significantly more likely to select the crowd and string analogies than the other two groups (p<0.05).Thus, we find a strong association between the representation presented to students and students' choices of analogy.
We next analyze student answers to questions 2 and 3, first according to experimental condition, and then according to the analogies selected by students on question 1.We find significant effects due to both the representations presented to students, and student use of analogy.
Figure 11.Percent of students in the abstract, concrete, and blend groups choosing single analogies (football, slinky, crowd, string) or two analogies (football/slinky, crowd/string).Other combinations accounted for less than 8% of student responses.

Question 2
Figure 12 shows student answers to question 2 according to treatment group, with substantial differences between the three groups.Here, we show the four main answers, accounting for 86% of student responses.On the correct answer (1=2=3>4), the abstract group was outperformed by both the blend (p=0.002) and concrete (p=0.024)groups, with the blend group demonstrating the highest performance of the three groups.Turning to the distracters, students in the abstract group were most likely to select 1>2=4>3 (p<0.002),followed second by students in the blend group (p<0.002), with students in the concrete group least likely to select this distracter.Students in the concrete group were most likely to select two other distracters, 1=2>4>3 (p<0.002) or 1=2>4=3 (p<0.002).

Question 3
Figure 13 shows student answers to question 3 according to experimental group.Here, we present the three main answers, accounting for 93% of student responses.On the correct answer ("left/right"), the concrete group outperformed the abstract (p=0.07)group, but performance by the blend and concrete groups were not significantly different (p=0.1).On the distracters, the abstract group was more likely than the concrete (p=0.005) and blend (p=0.064)groups to select "up/down".While we see differences, the effects of changes in representation on student answers to question 3 are limited, (p>0.05comparing treatment groups on the correct answer).
Given the marked effects of representation on question 2, it is noteworthy that we found student performance on question 3 depending only weakly on representation.Notably, the representations presented to students on the first two questions are absent on question 3. To gain some insight, we look within each treatment group to examine how students' analogies (as they selected in question 1) affected their reasoning.Table 4 shows the number of students (N) selecting a given analogy (single or multiple) on question 1 above the corresponding analogy, and the fraction of student answers to question 3 below.These associations between analogy and answer are all statistically significant (χ 2 , p<0.001).In the concrete and blend groups, the majority of students selected slinky and/or football analogies, and more than half of these students answered question 3 correctly.Conversely, students in the abstract group tended to select string and/or crowd analogies to a greater degree than student in the other groups.We found that among these students in the abstract group who selected string and/or crowd analogies, 51% answered "up/down" on question 3.However, a substantial number of students in the abstract group did select slinky and/or football analogies, and, within this select group, 63% answered question 3 correctly.Interestingly, we found that among the few students who selected only the football analogy, 55% of these students answered "to the right" on question 3 -far more than for any other analogy, and the only group choosing "right" to a considerable degree.

Discussion of the Within-Domain Study
The purpose of the within-domain study was to examine student reasoning within a single layer, sound waves.According to the analogical scaffolding model, signs can cue productive schemata for students to apply across multiple layers, and in the across-domain study one of these layers involved sound waves.The within-domain study demonstrated three key findings: (1) Signs can drive students' choice of analogy on question 1. (2) Students in the blend group productively applied the 3D idea to an abstract (sine wave) representation of sound on question 2, while students in the other treatments did not.(3) Absent an overt sign on question 3, there is only weak association between students' answers and the representations presented on earlier questions.However, we do find a stronger association between students' answers and the analogies they bear in mind.
Thus, representation can drive analogy and, therefore, schemata.We could represent this as an arrow pointing from sign to schema in Figures 4 and 5, indicating the direction of cueing.
[53] (Note that the slinky was the most popular choice in all three treatment groups, and, therefore, representation is one of several mechanisms driving analogy.)Within the sound waves layer, schemata preferentially cued by abstract and concrete signs were consistent with the predictions of the analogical scaffolding model.For instance, students presented with only an abstract sine wave on the quiz select answers to question 2 reflecting "up means up" reasoning about this representation of a sound wave (i.e., these students tended to select 1>2=4>3 in Figure 12).Conversely, students presented with a concrete picture of air particles selected answers reflecting 3D conceptions of a sound wave.However, note that in the concrete treatment, sound was represented by a concrete picture of air particles on question 2. We might therefore argue that students in the concrete group were able to map this information directly from the diagram shown in Figure 9.Only students in the blend group interpreted a sine wave as representing a 3D sound wave.According to the analogical scaffolding model, for the blend group, this sine wave took this 3D meaning by way of a prior blend (in question 1) with a concrete picture of air particles.This result confirms the models prediction that schemata that are tightly coupled to concrete signs preferentially project to blends over schemata (weakly) coupled to abstract signs.
[5] If concrete signs were not privileged in this way, we would expect many more students in the blend group to answer similarly to students in the abstract group (on both questions 2 and 3).
Without explicit signs, students' mechanistic reasoning about sound (i.e., motion of air particles) remains strongly coupled to the analogies they bear in mind, as evidenced by Table 4.
Interestingly, though students in the abstract group were more likely than other students to use "up means up" reasoning on question 3 (answering "up and down"), the three groups were not significantly different on the correct answer.We may describe this as "weak cueing", whereby the schemata coupled to questions 1 and 2 of the quiz were not strongly cued by the representation (or lack thereof) on question 3.In this situation, students may rely on the analogies they bear in mind (Table 4), or on other prior knowledge of sound.In summary, we note that one indicator of difficulty for students on questions 2 and 3 may be the use of representations.We find students in the abstract group relatively unprepared to interpret abstract representations on question 2, while students in the blend group demonstrated the highest level of ability to productively interpret these abstract representations.

Conclusion
As part of ongoing studies of student learning with analogy, we have conducted two sets of empirical studies to examine the utility of the analogical scaffolding model.In the first of these studies, we found that analogical scaffolding constitutes a productive tool for analyzing student learning with analogies.In this across-domain study, students taught about EM waves with a tutorial incorporating blends (appropriately presented according to the analogical scaffolding model) outperformed students taught the same material without blends.Students taught with blends achieved post-test scores three times those of students taught with canonical (abstract) representations alone.In addition to predicting which curricular materials are optimal (of the three used in the across-domain study), the model also explains why tutorials that did not use blends were less beneficial for student learning.Abstract signs (e.g., sine wave) do not always couple to productive schemata, while concrete signs (e.g., air particles) that are coupled to productive schemata do not readily cue these schemata across layers.However, abstract signs do cue productive schemata across layers when blended with a concrete sign in previous layers.
In a second complementary study, we examined student reasoning about sound waves, demonstrating how blends occur in particular instances.We find that signs can cue particular schemata and associated analogies that appear to drive student reasoning about sound waves.
Consistent with the across-domain study, we find abstract signs can couple to unproductive schemata when used alone, but these abstract signs can cue productive schemata when blended previously with a concrete sign.On a quiz focusing on sound waves, students presented with blends outperformed (by a factor of two) students presented with abstract representations alone in their ability to productively interpret these canonical representations.
These across-and within-domain studies provide consistent evidence in support of the weak hypothesis that signs can cue associated, but pre-existing, schemata.This cueing leads to significant variations in student reasoning about waves as measured by the assessments in both studies.Further, the across-domain study provides evidence in support of the strong hypothesis that signs and blending can lead to the formation of new schemata.The various ways these new schemata are formed may depend strongly on the signs used to teach.The across-and withindomain studies support the model of analogical scaffolding and provide a prototype for future studies of this kind.
Prull for supporting this work, Edward Redish, Thomas Bing, Michelle Zandieh, Michael Wittmann, and the PER at Colorado Group, particularly Michael Dubson and Patrick Kohl, for essential and significant contributions to this work.We also thank the students for their participation.

Figure 1 .
Figure 1.A sound wave represented by a picture of compressed and rarefied air particles (a) and a sine wave (b).

Figure 2 .
Figure 2. Questions given on the pre-and post-tests in the across-domain study.

Figure 3 .
Figure3.Examples of representations of a wave on a string, sound wave, and EM wave used on the abstract, concrete, and blend tutorials in the across-domain study.The complete set of representations can be found in the supplementary materials.[39]

Figure 4 .
Figure 4. Analogical scaffolding schematics for the abstract (left) and concrete (right) tutorials.

Figure 5 .
Figure 5. Analogical scaffolding schematic for the blend tutorial.Dashed lines delineate string, sound, and EM wave domains.

Figure 6 .
Figure 6.Fraction of correct answers on pre-post question 1 from the across-domain study.Error bars represent ± the standard error of the mean.

Figure 7 .
Figure 7. Fraction of correct (P=Q=R=S) and partially correct (P=Q=R>S) answers on pre-post question 2 from the across domain study.

Figure 9 .
Figure 9. Question 2 sound waves quiz.The same representation (top) was used for the abstract and blend groups.A different representation (bottom) was used for the concrete group.

Figure 8 .
Figure 8. Question 1 from the sound waves quiz.Each group (abstract, concrete, and blend) received a different representation shown on the right.

Figure 10 .
Figure 10.Question 3 from the sound waves quiz, identical for all three experimental groups.

Figure 12 .
Figure 12.Student answers to the sound waves quiz question 2 according to experimental group.The correct answer is the left-most dark gray bar (1=2=3>4).

Figure 13 .
Figure 13.Students answers to the sound waves quiz question 3 according to experimental group.The correct answer is the left-most dark gray bar (left/right).

Table 1 .
Across-domain Study Experimental Groups

Table 2 .
Coded student explanations on question 1 from the across-domain study pre-and post-tests.On the pre-test, and across all three groups on the post-test, we find similar patterns in

Table 2 .
[47]results are generally diagonalized, suggesting a strong association between answer and reasoning.[47]Studentsanswering correctly (I=J=K=L) used Same X Position reasoning, students answering I>J>K>L used Read as Graph reasoning, and students answering J>I=K>L used Proximity to Line reasoning.Zero students in the concrete group answered J>I=K>L on the post-test.Notably, the fraction of students using Sound Words increased from zero on the pre-test

Table 3 .
Within-domain study Experimental Groups

Table 4 .
Student answers to question 3 from the sound quiz, split by treatment and analogies selected.