arxiv: 2604.08693 · v1 · submitted 2026-04-09 · 💻 cs.CY · cs.HC· cs.IR

Recognition: unknown

Towards Generalizable Representations of Mathematical Strategies

Siddhartha Pradhan , Ethan Prihar , Erin Ottmar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:58 UTC · model grok-4.3

classification 💻 cs.CY cs.HCcs.IR

keywords algebraic solution pathwaysstrategy embeddingscontrastive learninglearning outcomesmathematical strategieseducational data miningproblem-invariant representations

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The pith

Subtracting vector representations of consecutive algebra steps produces embeddings that capture student strategies independently of the problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to represent entire algebraic solution paths in a way that stays the same across different problems. It does this by first taking the difference between the model encodings of one step and the next, then applying contrastive learning to pull similar strategies together and push dissimilar ones apart. These representations are tested on tasks like predicting the actions used in a solution and how efficient the path is. The authors also extract measures of how unique or conventional a student's approach is and show that these measures relate to both immediate performance and longer-term learning results. The goal is to replace manual labeling or platform-specific tracking with scalable, general indicators of mathematical thinking.

Core claim

By computing vector differences between consecutive algebraic states from pretrained encoders and then applying SimCSE contrastive objectives to entire sequences, the resulting embeddings encode meaningful strategy information that supports multi-label action classification, solution efficiency prediction, sequence reconstruction, and embedding-derived metrics of uniqueness, diversity, and conformity that correlate with short-term and distal learning outcomes.

What carries the argument

Transition embeddings formed by subtracting the vector encodings of one algebraic state from the next, followed by contrastive sequence-level embeddings that group paths by strategy similarity.

If this is right

The embeddings support automatic classification of solution actions without hand-labeled data.
Efficiency of a solution path can be predicted directly from its embedding.
Measures of strategy uniqueness and diversity serve as scalable indicators of divergent thinking.
Analyses of student behavior become possible across different problems and platforms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Tutoring systems could use these embeddings to detect when a student is using an uncommon but effective approach and adjust feedback accordingly.
Large-scale datasets from multiple curricula could be compared for patterns in how strategy diversity relates to long-term retention.
The same transition-and-contrastive approach might be tested on step-by-step solutions in physics or geometry to check whether the method generalizes beyond algebra.

Load-bearing premise

Subtracting consecutive state vectors from a pretrained model removes problem-specific content and leaves only the transformation that defines the strategy.

What would settle it

Collect a fresh set of student algebra solutions, compute the same embedding-based uniqueness and conformity scores, and find that they show no reliable statistical association with measured learning gains or creativity indicators.

Figures

Figures reproduced from arXiv: 2604.08693 by Erin Ottmar, Ethan Prihar, Siddhartha Pradhan.

**Figure 3.** Figure 3: Illustration of the SimCSE-based encoder model, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: t-SNE projections of individual state- and transition-level embeddings. (a) State-based embeddings colored by problem [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

read the original abstract

Pretrained encoders for mathematical texts have achieved significant improvements on various tasks such as formula classification and information retrieval. Yet they remain limited in representing and capturing student strategies for entire solution pathways. Previously, this has been accomplished either through labor-intensive manual labeling, which does not scale, or by learning representations tied to platform-specific actions, which limits generalizability. In this work, we present a novel approach for learning problem-invariant representations of entire algebraic solution pathways. We first construct transition embeddings by computing vector differences between consecutive algebraic states encoded by high-capacity pretrained models, emphasizing transformations rather than problem-specific features. Sequence-level embeddings are then learned via SimCSE, using contrastive objectives to position semantically similar solution pathways close in embedding space while separating dissimilar strategies. We evaluate these embeddings through multiple tasks, including multi-label action classification, solution efficiency prediction, and sequence reconstruction, and demonstrate their capacity to encode meaningful strategy information. Furthermore, we derive embedding-based measures of strategy uniqueness, diversity, and conformity that correlate with both short-term and distal learning outcomes, providing scalable proxies for mathematical creativity and divergent thinking. This approach facilitates platform-agnostic and cross-problem analyses of student problem-solving behaviors, demonstrating the effectiveness of transition-based sequence embeddings for educational data mining and automated assessment.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proposes transition embeddings from pretrained math encoders plus SimCSE to create problem-invariant representations of algebraic solution strategies, but the subtraction step lacks evidence it actually isolates transformations.

read the letter

The main thing here is a pipeline that subtracts consecutive state embeddings from high-capacity pretrained models to form transition vectors, then feeds those into SimCSE for sequence-level embeddings that are meant to be invariant across algebra problems. From there they extract measures of strategy uniqueness, diversity, and conformity and check correlations with learning outcomes. This is new compared to the manual labeling or platform-specific action logs mentioned in the abstract, and it targets a clear practical gap in scaling up analysis of student pathways without tying everything to one system. The attempt to link embedding-derived proxies to both short-term and longer-term outcomes is a reasonable direction for educational data mining. The soft spot is the central assumption that vector differences will cancel problem-specific features and highlight moves. Pretrained encoders are typically trained on static expressions or retrieval, not on differences, so nothing in the geometry ensures clean disentanglement. If problem identity leaks through, the SimCSE clusters and the derived measures could reflect problem similarity more than strategy, which would undermine the claimed correlations. The abstract gives no dataset details, baselines, statistical tests, or controls for confounds, so it is hard to judge how well the evaluations actually support the claims. This is for people working on learning analytics and automated assessment tools. A reader focused on scalable proxies for mathematical creativity or divergent thinking could pick up useful ideas on embedding sequences, but they would need to see the full methods and results to assess reliability. It deserves a serious referee because the framing is honest about prior limitations and the technical steps build on established embedding methods, even if the invariance claim needs stronger checks.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel approach to learn problem-invariant representations of entire algebraic solution pathways. Transition embeddings are formed by vector subtraction of consecutive algebraic states encoded by high-capacity pretrained models; sequence embeddings are then obtained via SimCSE contrastive learning. These representations are evaluated on multi-label action classification, solution efficiency prediction, and sequence reconstruction tasks, and are used to derive embedding-based measures of strategy uniqueness, diversity, and conformity that are reported to correlate with short-term and distal learning outcomes, serving as scalable proxies for mathematical creativity.

Significance. If the transition embeddings successfully isolate strategy transformations from problem identity, the work would supply a generalizable, platform-agnostic method for automated analysis of student algebraic strategies, reducing reliance on manual labeling and enabling cross-problem studies of divergent thinking and learning outcomes. The use of pretrained encoders plus contrastive sequence modeling is a creative extension of existing tools to educational data mining.

major comments (2)

[Transition embeddings construction] The central technical claim—that vector differences between consecutive states from pretrained encoders cancel problem-specific features while retaining transformations—is load-bearing for all downstream results yet receives no direct validation. No ablation, invariance test (e.g., same strategy on different base problems), or quantitative check that residual problem identity is negligible appears in the method description or experiments; without such evidence the SimCSE embeddings and the derived uniqueness/diversity/conformity measures risk capturing problem similarity rather than strategy.
[Evaluation and results] The evaluation section reports correlations between the derived measures and learning outcomes but provides no information on dataset sizes, problem distributions, statistical controls for student ability or problem difficulty, baseline comparisons, or significance testing. These omissions prevent assessment of whether the reported correlations are robust or could be explained by confounds.

minor comments (2)

Clarify the exact pretrained encoders employed, the construction of positive/negative pairs for SimCSE, and whether any fine-tuning occurs.
Add a limitations paragraph discussing potential failure modes of the subtraction step and the generalizability of the pretrained models to student-generated algebraic expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: The central technical claim—that vector differences between consecutive states from pretrained encoders cancel problem-specific features while retaining transformations—is load-bearing for all downstream results yet receives no direct validation. No ablation, invariance test (e.g., same strategy on different base problems), or quantitative check that residual problem identity is negligible appears in the method description or experiments; without such evidence the SimCSE embeddings and the derived uniqueness/diversity/conformity measures risk capturing problem similarity rather than strategy.

Authors: We agree that direct validation of the invariance property would strengthen the central claim. While the downstream tasks provide indirect support that the embeddings encode strategy information, we acknowledge the absence of explicit ablations or invariance tests. In the revised manuscript, we will include an ablation study comparing transition embeddings to raw state embeddings and an invariance test applying the same strategies to different base problems to quantify residual problem identity. revision: yes
Referee: The evaluation section reports correlations between the derived measures and learning outcomes but provides no information on dataset sizes, problem distributions, statistical controls for student ability or problem difficulty, baseline comparisons, or significance testing. These omissions prevent assessment of whether the reported correlations are robust or could be explained by confounds.

Authors: We thank the referee for highlighting these omissions. The original manuscript lacked details on dataset sizes, problem distributions, statistical controls, baseline comparisons, and significance testing. We will revise the evaluation section to include dataset statistics, describe the problem distributions and statistical controls for student ability and problem difficulty, add baseline comparisons, and report significance testing for the correlations to demonstrate their robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external pretrained models and established SimCSE without reduction to inputs by construction

full rationale

The paper constructs transition embeddings via vector subtraction on outputs from high-capacity external pretrained encoders, then applies the standard SimCSE contrastive objective to obtain sequence embeddings. The uniqueness, diversity, and conformity measures are computed directly from these embeddings and evaluated against independent learning outcome data. No step defines a target quantity in terms of itself, renames a fitted parameter as a prediction, or relies on self-citation chains for load-bearing justification. The central claims rest on the empirical performance of the combined pipeline rather than tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, the paper introduces no new free parameters, axioms, or invented entities; it builds on existing pretrained encoders and the SimCSE contrastive framework.

pith-pipeline@v0.9.0 · 5521 in / 1146 out tokens · 60320 ms · 2026-05-10T16:58:30.873091+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

75 extracted references · 9 canonical work pages · 2 internal anchors

[1]

Towards Generalizable Representations of Mathematical Strategies

INTRODUCTION Mathematical proficiency is an essential requirement for STEM disciplines and subsequent success in the STEM workforce[14]. Algebra, in particular, is considered foundational for learn- ing more advanced topics in mathematics [49]. However, many middle school and high school students struggle with basic algebraic concepts, including determini...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

DenominatorFactor

RELA TED WORK Learning representations for the entire solution pathways implies summarizing the employed strategies and all math- ematical transitions. In prior work, Ritter et al. [43] dis- cuss different approaches used to identify student strategies when solving math problems. Specifically, they define strat- egy as a sequence of steps or transitions t...
[3]

CURRENT WORK Taken together, while numerous advancements have been made in prior work, there is a need to identify new methods for characterizing and analyzing students’ complete solution pathways when solving mathematics problems. To address these gaps, we explore the following three research questions: RQ 1How do transition-based embeddings compare to s...
[4]

BACKGROUND 4.1 Facets of Algebraic Learning Algebraic proficiency requires not only procedural and con- ceptual knowledge, but also the ability to apply this knowl- edge flexibly [47]. Procedural knowledge is defined as the ordering of algebraic steps and transitions required to solve a problem, whereas conceptual understanding relates to the knowledge of...
[5]

11 + 55 +y+ 89 + 45

METHODS 5.1 Data and Context In this section, we describe the overall context of the study, including the learning platform from which the data was obtained and its source. 5.1.1 From Here To There! (FH2T) FH2T is a gamified learning platform grounded in principles of perceptual learning, embodied cognition, and gamification [36]. The design of FH2T empha...

2020
[6]

Transi- tion Embeddings Figure 4 visualizes the geometric structure ofindividualstate- and transition-based embeddings using t-SNE projections

RESULTS 6.1 RQ1: Generalizability of State vs. Transi- tion Embeddings Figure 4 visualizes the geometric structure ofindividualstate- and transition-based embeddings using t-SNE projections. Each point corresponds to a single algebraic state or tran- sition. For state-based embeddings (Figure 4a), the embed- dings form well-separated clusters that align c...
[7]

DISCUSSION A core objective of Educational Data Mining and Learning Analytics is to develop scalable and generalizable representa- tions that capture student learning behaviors and strategies across tasks and platforms. This study contributes to this goal by examining whether sequence-level representations of student solution pathways can capture meaningf...
[8]

First, we considered a limited set of base en- coders, which restricts the generality of our findings across substantially different model architectures

LIMITA TIONS While our approach enables strategy-level analysis that gen- eralizes across problems, its applicability is constrained by several factors. First, we considered a limited set of base en- coders, which restricts the generality of our findings across substantially different model architectures. Additionally, the inclusion of domain-specific enc...
[9]

FUTURE DIRECTIONS This work opens several directions for future research for enhancing teaching and learning. One promising application is the development of systems that automatically suggest alternative solution approaches that are maximally different from a student’s current strategy, encouraging exploration and divergent thinking. More refined and the...
[10]

CONCLUSION In this study, we introduced a method for learning represen- tations of entire algebraic solution pathways. By leveraging pretrained mathematical encoders and contrastive sequence- level learning via SimCSE, we showed that transition-based embeddings capture the structural and strategic characteris- tics of student solutions. Evaluation across ...
[11]

Department of Education, through an Efficacy and Replication Grant (R305A180401) and an NSF CAREER Grant (2142984) to Worcester Polytechnic Institute

ACKNOWLEDGMENTS The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through an Efficacy and Replication Grant (R305A180401) and an NSF CAREER Grant (2142984) to Worcester Polytechnic Institute. The opinions expressed are those of the authors and do not represent the views of the Institute or the U...
[12]

Y. Adi, E. Kermany, Y. Belinkov, O. Lavi, and Y. Goldberg. Fine-grained Analysis of Sentence Embeddings Using Auxiliary Prediction Tasks, Feb
[13]

arXiv:1608.04207 [cs]

work page Pith review arXiv
[14]

Agirre, C

E. Agirre, C. Banea, D. Cer, M. Diab, A. Gonzalez-Agirre, R. Mihalcea, G. Rigau, and J. Wiebe. Semeval-2016 task 1: Semantic textual similarity, monolingual and cross-lingual evaluation. In Proceedings of the 10th international workshop on semantic evaluation (SemEval-2016), pages 497–511, 2016

2016
[15]

Alfieri, T

L. Alfieri, T. J. Nokes-Malach, and C. D. Schunn. Learning through case comparisons: A meta-analytic review.Educational Psychologist, 48(2):87–113, 2013

2013
[16]

R. S. Baker, A. T. Corbett, and A. Z. Wagner. Human classification of low-fidelity replays of student actions. InProceedings of the educational data mining workshop at the 8th international conference on intelligent tutoring systems, volume 2002, pages 29–36, 2006

2002
[17]

R. E. Beaty and D. R. Johnson. Automating creativity assessment with SemDis: An open platform for computing semantic distance.Behavior Research Methods, 53(2):757–780, 2021

2021
[18]

Bordes, N

A. Bordes, N. Usunier, A. Garcia-Duran, J. Weston, and O. Yakhnenko. Translating Embeddings for Modeling Multi-relational Data. InAdvances in Neural Information Processing Systems, volume 26. Curran Associates, Inc., 2013

2013
[19]

J. Y.-C. Chan, E. R. Ottmar, and J.-E. Lee. Slow down to speed up: Longer pause time before solving problems relates to higher strategy efficiency.Learning and Individual Differences, 93:102109, 2022

2022
[20]

Conneau and D

A. Conneau and D. Kiela. SentEval: An Evaluation Toolkit for Universal Sentence Representations, Mar. 2018

2018
[21]

L. E. Decker-Woodrow, C. A. Mason, J.-E. Lee, J. Y.-C. Chan, A. Sales, A. Liu, and S. Tu. The impacts of three educational technologies on algebraic understanding in the context of COVID-19.AERA open, 9:23328584231165919, 2023. Publisher: SAGE Publications Sage CA: Los Angeles, CA

2023
[22]

R. L. DeHaan. Teaching creativity and inventive problem solving in science.CBE life sciences education, 8(3):172–181, 2009

2009
[23]

Drechsel, A

J. Drechsel, A. Reusch, and S. Herbold. MAMUT: A Novel Framework for Modifying Mathematical Formulas for the Generation of Specialized Datasets for Language Model Training, July 2025. arXiv:2502.20855 [cs]

work page arXiv 2025
[24]

M. Q. Feldman, J. Y. Cho, M. Ong, S. Gulwani, Z. Popovi´ c, and E. Andersen. Automatic Diagnosis of Students’ Misconceptions in K-8 Mathematics. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems, CHI ’18, pages 1–12, New York, NY, USA, Apr. 2018. Association for Computing Machinery

2018
[25]

T. Gao, X. Yao, and D. Chen. SimCSE: Simple Contrastive Learning of Sentence Embeddings. In M.-F. Moens, X. Huang, L. Specia, and S. W.-t. Yih, editors,Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 6894–6910, Online and Punta Cana, Dominican Republic, Nov. 2021. Association for Computational Linguistics

2021
[26]

Grover and R

S. Grover and R. Pea. Computational Thinking in K–12: A Review of the State of the Field.Educational Researcher, 42(1):38–43, Jan. 2013

2013
[27]

D. W. Haylock. Mathematical creativity in schoolchildren.The Journal of Creative Behavior, 21(1):48–59, 1987

1987
[28]

Gaussian Error Linear Units (GELUs)

D. Hendrycks and K. Gimpel. Gaussian Error Linear Units (GELUs), June 2023. arXiv:1606.08415 [cs]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[29]

F. Hill, K. Cho, and A. Korhonen. Learning Distributed Representations of Sentences from Unlabelled Data. In K. Knight, A. Nenkova, and O. Rambow, editors,Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 1367–1377, San Diego, California, June 2016. Associa...

2016
[30]

Hochreiter and J

S. Hochreiter and J. Schmidhuber. Long Short-Term Memory.Neural Computation, 9(8):1735–1780, Nov. 1997

1997
[31]

W. Hong, J. R. Star, R.-D. Liu, R. Jiang, and X. Fu. A Systematic Review of Mathematical Flexibility: Concepts, Measurements, and Related Research. Educational Psychology Review, 35(4):104, Dec. 2023

2023
[32]

N. N. Kennard, G. Angeli, and C. D. Manning. Evaluating word embeddings using a representative suite of practical tasks. InProceedings of the 1st workshop on evaluating vector-space representations for nlp, pages 19–23, 2016

2016
[33]

E. J. Knuth, A. C. Stephens, N. M. McNeil, and M. W. Alibali. Does understanding the equal sign matter? Evidence from solving equations.Journal for research in Mathematics Education, 37(4):297–312, 2006

2006
[34]

K. R. Koedinger and M. J. Nathan. The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning.Journal of the Learning Sciences, 13(2):129–164, Apr. 2004. eprint: https://doi.org/10.1207/s15327809jls1302 1

work page doi:10.1207/s15327809jls1302 2004
[35]

J. Kolb, S. Farrar, and Z. A. Pardos. Generalizing Expert Misconception Diagnoses through Common Wrong Answer Embedding. Technical report, International Educational Data Mining Society, July
[36]

ERIC Number: ED599212
[37]

O. N. Kwon, J. H. Park, and J. S. Park. Cultivating divergent thinking in mathematics through an open-ended approach.Asia Pacific Education Review, 7(1):51–61, July 2006

2006
[38]

J.-E. Lee, J. Y.-C. Chan, A. Botelho, and E. Ottmar. Does slow and steady win the race?: Clustering patterns of students’ behaviors in an interactive online mathematics game.Educational technology research and development, 70(5):1575–1599, 2022

2022
[39]

J.-E. Lee, C. B. Hornburg, J. Y.-C. Chan, and E. Ottmar. Perceptual and Number Effects on Students’ Initial Solution Strategies in an Interactive Online Mathematics Game.Journal of Numerical Cognition, 8(1):166–182, Mar. 2022

2022
[40]

R. Leikin. Exploring Mathematical Creativity Using Multiple Solution Tasks. InCreativity in Mathematics and the Education of Gifted Students, pages 129–145. Brill, Jan. 2009. Section: Creativity in Mathematics and the Education of Gifted Students

2009
[41]

Leikin and M

R. Leikin and M. Lev. Multiple solution tasks as a magnifying glass for observation of mathematical creativity. InProceedings of the 31st international conference for the psychology of mathematics education, volume 3, pages 161–168, 2007

2007
[42]

H. Li, M. Xu, and Y. Song. Sentence Embedding Leaks More Information than You Expect: Generative Embedding Inversion Attack to Recover the Whole Sentence. In A. Rogers, J. Boyd-Graber, and N. Okazaki, editors,Findings of the Association for Computational Linguistics: ACL 2023, pages 14022–14040, Toronto, Canada, July 2023. Association for Computational Li...

2023
[43]

Logeswaran and H

L. Logeswaran and H. Lee. An efficient framework for learning sentence representations. Feb. 2018

2018
[44]

X. Ma, Z. Wang, P. Ng, R. Nallapati, and B. Xiang. Universal Text Representation from BERT: An Empirical Study, Oct. 2019. arXiv:1910.07973 [cs]

work page arXiv 2019
[45]

L. v. d. Maaten and G. Hinton. Visualizing data using t-SNE.Journal of machine learning research, 9(Nov):2579–2605, 2008

2008
[46]

J. X. Morris, V. Kuleshov, V. Shmatikov, and A. M. Rush. Text Embeddings Reveal (Almost) As Much As Text, Oct. 2023. arXiv:2310.06816 [cs]

work page arXiv 2023
[47]

Ottmar, D

E. Ottmar, D. Landy, and R. Goldstone. Teaching the perceptual structure of algebraic expressions: Preliminary findings from the pushing symbols intervention. InProceedings of the Annual Meeting of the Cognitive Science Society, volume 34, 2012

2012
[48]

Ottmar, J.-E

E. Ottmar, J.-E. Lee, K. Vanacore, S. Pradhan, L. Decker-Woodrow, and C. A. Mason. Data from the efficacy study of from here to there! A dynamic technology for improving algebraic understanding. Journal of Open Psychology Data, 11(1), 2023

2023
[49]

E. R. Ottmar, D. Landy, R. Goldstone, and E. Weitnauer. Getting From Here to There! : Testing the Effectiveness of an Interactive Mathematics Intervention Embedding Perceptual Learning. Proceedings of the Annual Meeting of the Cognitive Science Society, 37(0), 2015

2015
[50]

Paquette, A

L. Paquette, A. de Carvahlo, R. Baker, and J. Ocumpaugh. Reengineering the feature distillation process: A case study in detection of gaming the system. InEducational data mining 2014, 2014

2014
[51]

Pradhan, A

S. Pradhan, A. Gurung, and E. Ottmar. Gamification and deadending: Unpacking performance impacts in algebraic learning. InProceedings of the 14th learning analytics and knowledge conference, pages 899–906, 2024

2024
[52]

Pradhan, E

S. Pradhan, E. Ottmar, A. Gurung, and J.-E. Lee. MathFlowLens: a classification and visualization tool for analyzing students’ procedural pathways: S. Pradhan et al.Educational technology research and development, pages 1–26, 2025

2025
[53]

Reimers and I

N. Reimers and I. Gurevych. Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks. In K. Inui, J. Jiang, V. Ng, and X. Wan, editors, Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 3982–3992, Hong Kong, China, Nov...

2019
[54]

Reusch, J

A. Reusch, J. Gonsior, C. Hartmann, and W. Lehner. Investigating the Usage of Formulae in Mathematical Answer Retrieval. InAdvances in Information Retrieval: 46th European Conference on Information Retrieval, ECIR 2024, Glasgow, UK, March 24–28, 2024, Proceedings, Part I, pages 247–261, Berlin, Heidelberg, Mar. 2024. Springer-Verlag

2024
[55]

Reusch, M

A. Reusch, M. Thiele, and W. Lehner. Transformer-encoder and decoder models for questions on math. 2022. Authority: CLEF

2022
[56]

Ritter, R

S. Ritter, R. Baker, V. Rus, and G. Biswas. Identifying strategies in student problem solving. Design Recommendations for Intelligent Tutoring Systems, 7:59–70, 2019

2019
[57]

Rittle-Johnson, M

B. Rittle-Johnson, M. Schneider, and J. R. Star. Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics.Educational Psychology Review, 27(4):587–597, Dec. 2015

2015
[58]

Rittle-Johnson, R

B. Rittle-Johnson, R. S. Siegler, and M. W. Alibali. Developing conceptual understanding and procedural skill in mathematics: An iterative process.Journal of educational psychology, 93(2):346, 2001

2001
[59]

Rittle-Johnson, J

B. Rittle-Johnson, J. R. Star, and K. Durkin. Developing procedural flexibility: Are novices prepared to learn from comparing procedures?British Journal of Educational Psychology, 82(3):436–455, Sept. 2012

2012
[60]

Schneider, B

M. Schneider, B. Rittle-Johnson, and J. R. Star. Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge.Developmental psychology, 47(6):1525, 2011

2011
[61]

A. H. Schoenfeld. Problem solving in the United States, 1970–2008: research and theory, practice and politics.ZDM, 39(5-6):537–551, Sept. 2007

1970
[62]

Selden and J

A. Selden and J. Selden. Reflections on foundations for success: The final report of the national mathematics advisory panel, 2009

2009
[63]

Shakya, V

A. Shakya, V. Rus, and D. Venugopal. Student Strategy Prediction Using a Neuro-Symbolic Approach. Technical report, International Educational Data Mining Society, 2021. ERIC Number: ED615630

2021
[64]

J. T. Shen, M. Yamashita, E. Prihar, N. Heffernan, X. Wu, B. Graff, and D. Lee. MathBERT: A Pre-trained Language Model for General NLP Tasks in Mathematics Education, Aug. 2023. arXiv:2106.07340 [cs]

work page arXiv 2023
[65]

Srivastava, G

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting.J. Mach. Learn. Res., 15(1):1929–1958, Jan. 2014

1929
[66]

J. D. Stanton, A. J. Sebesta, and J. Dunlosky. Fostering metacognition to support student learning and performance.CBE—Life Sciences Education, 20(2):fe3, 2021

2021
[67]

R. J. Sternberg and T. I. Lubart. The concept of creativity: Prospects and paradigms. InHandbook of creativity, pages 3–15. Cambridge University Press, New York, NY, US, 1999

1999
[68]

Tabach and A

M. Tabach and A. Friedlander. Algebraic procedures and creative thinking.ZDM, 49(1):53–63, Mar. 2017

2017
[69]

Thapa Magar, S

A. Thapa Magar, S. E. Fancsali, V. Rus, A. Murphy, S. Ritter, and D. Venugopal. Learning Representations for Math Strategies using BERT. InProceedings of the Eleventh ACM Conference on Learning @ Scale, L@S ’24, pages 514–518, New York, NY, USA, July 2024. Association for Computing Machinery

2024
[70]

Thapa Magar, A

A. Thapa Magar, A. Shakya, S. E. Fancsali, V. Rus, A. Murphy, S. Ritter, and D. Venugopal. ”Can A Language Model Represent Math Strategies?”: Learning Math Strategies from Big Data using BERT. InProceedings of the 15th International Learning Analytics and Knowledge Conference, LAK ’25, pages 655–666, New York, NY, USA, Mar. 2025. Association for Computing...

2025
[71]

Vaswani, N

A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need.Advances in neural information processing systems, 30, 2017

2017
[72]

Verschaffel, K

L. Verschaffel, K. Luwel, J. Torbeyns, and W. Van Dooren. Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education.European Journal of Psychology of Education, 24(3):335–359, Sept. 2009

2009
[73]

Z. Wang, M. Zhang, R. G. Baraniuk, and A. S. Lan. Scientific Formula Retrieval via Tree Embeddings. In 2021 IEEE International Conference on Big Data (Big Data), pages 1493–1503, Dec. 2021

2021
[74]

R. M. Welder. Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties.School Science and Mathematics, 112(4):255–264, 2012. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1949- 8594.2012.00136.x

work page doi:10.1111/j.1949- 2012
[75]

Zhang, Z

M. Zhang, Z. Wang, R. Baraniuk, and A. Lan. Math Operation Embeddings for Open-ended Solution Analysis and Feedback, Apr. 2021

2021