A causal design for long-term effects of Modified Moore Method instruction on baseline-strong students

A causal design for long-term effects of Modified Moore Method instruction on baseline-strong students

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

The Moore Method and modified versions use student work on mathematical problems and proof presentations; IBL is a broader family of inquiry-based approaches. Prior achievement identifies performance before the course, whereas later development is a distinct outcome. A learning trajectory describes changes in understanding or practice over time. Problem interweaving and student conjecturing are instructional choices.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied text contains quoted study discussion and several appended proposals, with unresolved choices of strong students, potential, outcomes, and measurement. Those statements are preserved as historical source material and are not asserted as current validated conclusions.

Catalogue screening limitation: Empirical education study and classroom-culture questions, rather than a mathematical statement.

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Statement recorded in the supplied source.

$1$ What are long term effects of MMM instruction upon strong students? $2$ How do these effects compare with those of more traditional, lecture-based instruction (and possibly more general IBL)? $3$ How can we further describe student development within the strong/high-achieving category? (see Tranformation Study) $4$ In what ways do MMM courses support this development, specifically for strong students?

Abstract: The following statement appears in the final report of the Colorado Study, Chapter 9: Summary of Findings 3

Overall, it appeared that non-IBL courses tended to reinforce prior achieve-ment patterns, helping the "rich" to get "richer." In contrast, IBL courses seemed to offer an extra boost to lower achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students. Indeed, high-achieving students may be encouraged by an IBL expe-rience to take more mathematics courses, especially more IBL courses $6.6.2$ again, consistent with instructor observations that strong students found the IBL approach stimulating $8.2.5$. The study did recognize and record different forms of IBL, primarily individual pre-sentation and group work. However, it went on to aggregate them under the label "IBL," in order to provide enough data to draw statistically significant conclusions. We applaud the findings concerning lower achieving students 4. However, we also rec-ognize that the strongest students of today will be the leaders of tomorrow, and that it is important to assist them in achieving their full potential. Doing them "no harm" does not seem likely to achieve that. We would like to investigate the possible benefits of IBL to these students mentioned in the above quote from the Colorado Study. The idea of this study is that these benefits might be more pronounced and easier to establish if it is focused on the benefits of the MMM, rather than IBL in general. 2Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. $2014$. Benefits for women and men of inquiry-based learning in college mathematics: A Multi-institution study. Journal for Research in Mathematics Education, 45$4$, 406418. 3S. Laursen, M. Hassi, M. Kogan, A. Hunter, Evaluation of the IBL Mathematics Project, Assessment & Evaluation Center for IBL in Mathematics, Ethnography and Evaluation Research and Tim Weston ATLAS Assessment and Research Center, University of Colorado at Boulder $2011$, p169. 4Laursen, et al., divided students into Low achieving (Math GPA 2.5 or below), High achieving (Math GPA 2.5 or above) categories. 7

The purpose of this study is to determine if there is validity to the commonly held belief by MMM instructors that, among students previously identified as strongest, MMM students tend subsequently to fulfill more of their potential than those taking the same courses in a traditional, lecture-based setting. To that end, we wish to more carefully examine the benefits of Modified Moore Method courses to strong students. By "strong" students we mean those students that are either hard-working and diligent, or talented and smart, or both. Below we propose some possible proxy measures for these traits. Thus far we have put together two possible approaches: $1$ We wish to compare the effect of MMM vs. lecture-based instruction upon strong students. $2$ We wish to examine high-achieving students (after becoming informed by our previous transformation study), and accomplish a similar goal of describing developmental categories at a higher resolution for this group of students. In order to investigate this pair of research questions, it will first be necessary to determine the following: $1$ How are strong incoming freshmen best identified? (Perhaps by high school grades? SAT grades? Other?) $2$ How is realization of potential by college graduates who declared a mathematics major best measured? (Perhaps by completion of a mathematics degree after declaring the major? By taking mathematics courses or doing individualized work in mathematics beyond the requirements for the degree? By acceptance to graduate school? By individual testimony? Other?) There are many ways achievement of potential could be manifest. The study will rely on identifying ways that are practical to measure. Once satisfactory metrics have been identified and described, the study may be carried out in the following steps: $1$ Institutions will be identified which have mathematics courses that are taught some-times by MMM and sometimes in a traditional, lecture-based format. $2$ For those institutions, records will be gathered of recent graduates who had at some point declared a mathematics major. $3$ From those records, students who qualify as "strong" will be selected. $4$ From the selected students, two groups will be formed. One group will consist of those who had taken the most MMM courses, and the other will consist of those who had taken the most lecture-based courses. $5$ Data on realization of potential will then be analyzed and used to compare MMM-students with lecture-based-students. The investigation of research questions 3 and 4 above will be informed by our planned "Exploratory Study of Student Development Under the Modified Moore Method". That study will describe student developmental categories and describe how students advance through these categories. We wish to build on that study by examining the highest-achieving 8

category in greater resolution. That is, how can we explain and describe student development in a MMM course within the highest-achieving category?

Problem sequences & learning trajectories. Intellectual cross-training.

Research Question: When interweaving content areas within a problem sequence, what are the impacts on students success, conceptual development, relationships between related concepts, their images of the roles of definitions and theorems, and equivalence of approaches?

Abstract: IBL Real Analysis instructors often interweave problems on different concepts rather than treating each concept independently. We propose to study whether doing so increases the odds of success for individual students and, if so, why. Perhaps having prob-lems spanning multiple concepts on which to work each evening allows students to make connections between these concepts thereby increasing the students' understanding of each concept as well as the interplay between them. What is an optimal amount of interweaving? We illustrate the phenomena we wish to study via two examples. $1$ An instructor might alternate, within a problem sequence, problems related to limit points and problems related to sequences prior to stating a problem that ties the two concepts together, such as every infinite bounded sequence has a limit point. $2$ An instructor might have problems intertwined between continuity and differentia-bility, with the concluding problem being to prove that every differentiable function is continuous.

Strategic Walls.

Research Question: How does an instructor create an adequate number of conjec-tures that will effectively demonstrate students need for mathematical need for proof in order to verify or invalidate their intuition? What are the features and timing of these conjectures for optimal effectiveness?

Abstract: Pedagogical conjecturing activities are intended to both develop students intu-itions about the content in real analysis and reinforce students understanding of the need for proof or counterexample. Real analysis is a particularly appropriate site for this question since students start with significant intuitions (from calculus) that is not necessarily consis-tent with the foundations of analysis (completeness, monsters, etc.). Specific types of these activities may include

Traps:: contradicts standard obvious intuitions

Hooks:: build new (perhaps surprising) intuitions, e.g., the cantor set

Generalizing:: asking whether a recently proven statement holds for a broader class of tasks What does implementing such strategies communicate to students about $1$ correct proof? $2$ their own responsibility for their learning? $3$ the instructor's responsibility for their learning? $4$ mathematical authority? $5$ what it means to "proof schemes? $2$ affect students" $6$ potentially, conceptual understanding? 9

$7$ workload? $8$ the difficulty of the concepts? $9$ the amount of coverage accomplished by the course? In an extreme version of this question, we seek to understand the differences in the devel-opment of students habits of mind when approaching mathematical tasks if the course is structured as statements, with many $1/4? 1/2?$ are false.

Beyond proof.

Research Question: What are the impacts of incorporating mathematical activity such as defining or conjecturing at different points or with different frequency in an IBL course?

Abstract: The classic Moore Method course involves the distribution of a sequence of def-initions and problems students are charged with solving. Other IBL strategies may ask students to generate the definitions and problems themselves. The choice of strategies on this continuum may depend on the instructors goal for the course. For example, requiring students to participate in the defining and conjecturing tasks may result in a different con-ception of the nature of mathematical inquiry and the culture of mathematics. On the other hand, it may be the case that requiring students to generate every definition in a course may discourage students or require an undesirable amount of time. We wish to study the optimal frequency, locations, and depth of these experiences, both defining and conjecturing. Many effects on student perception would be of interest, including perception of: student and in-structor responsibility for learning, mathematical authority, what it means to do math, their understanding, course difficulty, course coverage.

Designing in the Zone of Proximal Development.

Research Question: When teaching from a set of notes, how does an experienced IBL instructor identify when a problem sequence needs to be reconstructed to support the particular learning needs of their students, how do they accomplish it, and how do they assess its effectiveness?

Abstract: Course notes are more static than actual problem sequences implemented by IBL instructors. Effective instructors often modify problems in response to the ways that particular classes develop. For example a particular problem may be too difficult as stated and need to be broken into sub-steps to allow student success. Alternately, an instructor may realize that some of their students may be better served by using an alternate definition to a particular term, thus changing the trajectory of problems (e.g., the old definition now becomes a theorem). We seek to understand what aspects of students understanding an instructor must attend to in order effectively identify the need for these modifications and to understand the design principles that allow them to successfully modify the sequence. We also seek to understand any aspects of students or topics that may affect this process such as differences in when working with content for which students have a rich concept image vs. when a concept image is relatively absent. In addition to studying how this process 10

unfolds during a course, we may also explore how the standard versions of course notes may be modified outside of a particular implementation.

Proof. How does an IBL class impact students' understanding of proof?.

Research Questions: In what ways does an IBL class 5:$1$ impact students' proof schemes? $2$ affect students' understanding of the contextual meanings in mathematics of the words "axioms, definitions, conjecture, theorem" and their role in mathematics? $3$ help students develop competence at proof? What features of IBL create these impacts?

Abstract: Harel & Sowder have proposed a framework to classify and examine students' proof schemes. They have looked mostly at students in traditional courses, or as part of teaching experiments. We conjecture that in those implementations of IBL that provide experiences to students in which they must prove propositions on their own, students' proof schemes will improve, as will their understanding of proof, and their competence at writing proofs. We propose a qualitative study (a case study possibly) of a Real Analysis IBL course. Data will include documentation of the IBL implementation (syllabus, problem sequence, classroom observations, instructor's interview), evidence of students' procedural and concep-tual understanding of proof (artifacts such as proofs worked in tests or homework assign-ments, clinical interviews, preand post-surveys). Things to think about:

• If the study is done in two sites (e.g., Texas and India), how to ensure that classrooms observations are reliable and consistent? (Ideas: video, Skype interviews.)

• Why Real Analysis? It could be done in other courses, but RA is a good one: students have some experience with proofs, they are more mature mathematically, course content is challenging enough.

• Data should include students' demographic information, previous IBL experience, academic perfomance prior to the course.

• Teasing out what features of IBL cause change may be more difficult than gathering evidence of such change.

• We need to consider follow up evidence of the robustness of change or impact.

• Data collection instruments (interview protocols, proof tasks, etc.) will be piloted both in IBL and non-IBL courses, as possible.

• A more comprehensive literature review should be conducted to inform data collection and data analysis strategies, as well as useful frameworks to use or to adapt for this study. [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

Study long-term effects of Modified Moore Method instruction and the related learning-trajectory and proof proposals in the extract.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002206 (AIM-INFRASTRUCTURE-0106). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-INFRASTRUCTURE-0106. http://aimath.org/pastworkshops/iblanalysisrep.pdf.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20002206 and code AIM-INFRASTRUCTURE-0106 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The complete catalogue body above retains the original agenda, source status, and historical preparation. The complete primary IBL workshop report was read. Its adjacent study fragments remain in the reproduced source body; this attempt does not silently discard them. The catalogue's earlier proposed frameworks and research summaries are context, not established solutions credited to this attempt. The following is an explicitly limited investigation of one route; it neither substitutes a formalized toy model for the entire source agenda nor claims empirical or deployed-software results. Elementary auxiliary facts are not new original-target progress.

Testing a comparison restricted to later graduates

The long-term Modified Moore Method agenda needs a baseline definition of strong students and follow-up outcomes. Consider the attempted shortcut of comparing only students who eventually graduate. The following deliberately small causal model shows why post-instruction eligibility can invalidate even an initially randomized comparison. All modeled students are eligible at baseline. Let $M$ be a fair randomized instruction indicator, let $U$ be an independent fair binary baseline trait, and define outcome potential values $Y(0)=Y(1)=U$. Define graduation by $G=M\lor U$. These rules are hypothetical and make no assertion about real instruction or students. Thus the inference from an association within later graduates to a causal effect on baseline-strong students fails through outcome-related selection. Restricting baseline eligibility is a different operation from conditioning on a later event affected by instruction. The example does not show that this bias occurs in the report's data, which were not available here, or that a particular follow-up design eliminates every bias.

Verification and remaining gaps

The four equally likely $(M,U)$ states were enumerated, giving the stated conditional means and zero unit-level effects. No long-term cohort was studied, no strength measure was validated, and no benefit or harm of MMM was estimated. The several study proposals and appended material in the complete source remain unresolved. This familiar selection mechanism is auxiliary analysis only.

References and source attribution

Primary source: https://aimath.org/pastworkshops/iblanalysisrep.pdf. Read complete primary ten-page workshop report, including the classroom culture, proof-learning, and Modified Moore Method study proposals and appended material.

All retrievals were checked on 11 October 2026. Original contributor and catalogue attribution remains in the preserved Sources section. The elementary arguments below are self-contained; their exposition is not a claim of first discovery.

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Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

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