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arxiv: 1907.07749 · v1 · pith:BQRS3TSOnew · submitted 2019-07-17 · 🧮 math.HO

Student Inquiry and the Rascal Triangle

Philip K. Hotchkiss This is my paper

Pith reviewed 2026-05-24 19:56 UTC · model grok-4.3

classification 🧮 math.HO
keywords Rascal Triangleinquiry-based learningmathematics for liberal artsstudent discoveriescombinatorial patternspattern recognition
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The pith

Mathematics for liberal arts students can discover original patterns in the Rascal Triangle through inquiry-based exploration.

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

The paper describes an inquiry-based learning activity in which mathematics for liberal arts students examined the Rascal Triangle for patterns. Students brought their existing knowledge to the task and uncovered several patterns that appear to be new. This supports the idea that these students are capable of mathematical discovery when provided with appropriate structure and freedom. The example illustrates how inquiry-based approaches can reveal hidden student potential in courses often viewed as remedial.

Core claim

In an inquiry-based investigation, MLA students given the Rascal Triangle were able to look for and find patterns, resulting in some original discoveries not previously documented in the literature on the triangle.

What carries the argument

The Rascal Triangle, a combinatorial array on which students apply pattern-finding techniques during open exploration.

If this is right

  • Students in MLA courses can contribute original mathematical insights about combinatorial objects.
  • Inquiry-based learning structures enable non-majors to identify new relations in arrays like the Rascal Triangle.
  • The process allows students to apply prior knowledge directly to pattern recognition tasks.
  • Such activities demonstrate that courses for liberal arts students can produce new combinatorial observations.

Where Pith is reading between the lines

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

  • Similar open-ended inquiries on other combinatorial triangles could surface additional student-found relations.
  • The approach implies that verifying student patterns against existing literature becomes a necessary step in classroom discovery.
  • Broader adoption might change how introductory courses structure pattern-seeking tasks to include more student-driven elements.

Load-bearing premise

The patterns found by the students are genuinely new and not already known in the mathematical literature.

What would settle it

A search of published work on the Rascal Triangle that shows the same patterns were already described before the student activity.

Figures

Figures reproduced from arXiv: 1907.07749 by Philip K. Hotchkiss.

Figure 1
Figure 1. Figure 1: A triangular array. Instead of providing the row from Pascal’s Triangle that the instructor expected, 1 4 6 4 1 they produced the row 1 4 5 4 1. They did this by using the rule that the outside numbers are 1s and the inside numbers are determined by the diamond formula South = East · West + 1 North where North, South, East and West form a diamond in the triangular array as in [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 2
Figure 2. Figure 2: North, South, East and West entries in a triangular array. Continuing with this rule Anggaro, Liu and Tulloch created a number triangle they called the Rascal Triangle [1] [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Rascal Triangle [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Location of Entries The following definition was also helpful. Definition. For any number triangle, the diagonals running from right to left are called the major diagonals while the diagonals running from left to right are called the minor diagonals. (a) major diago￾nals. (b) Minor diago￾nals. Due to time constraints, proofs of the student patterns were not presented in class. However, they are accessible … view at source ↗
Figure 6
Figure 6. Figure 6: T-Meg Rule. For example, 16 = North + North0 + North1 = 9 + 1 + 6 Note that T-Meg’s Rule also holds for the second entry in a row as well. 8 = North + North0 + North1 = 1 + 1 + 6 As Meaghan wrote about her role in finding the T-Meg Rule (Example 1 below) The hockey still pattern in Pascal’s triangle is what helped me find the T-Meg pattern in Rascal’s triangle. I knew there was a similar hockey stick patte… view at source ↗
Figure 7
Figure 7. Figure 7: Ashley’s Rule [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: John’s Odd Diamond Pattern. 13 + 16 + 19 + 25 + 31 + 26 + 21 + 17 8 = 168 8 = 21 7+9+11+13+15+22+29+36+43+37+31+25+19+16+13+10 16 = 336 16 = 21 b. John’s Even Diamond Pattern: If you form a 2×2-diamond in the Rascal Triangle and a diamond with 2n numbers on each side that has the 2×2- diamond in the center, then the average of the 8n − 4 numbers along the edges of the outer diamond is equal to the average … view at source ↗
Figure 9
Figure 9. Figure 9: John’s Even Diamond pattern [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
read the original abstract

Those of us who teach Mathematics for Liberal Arts (MLA) courses often underestimate the mathematical abilities of the students enrolled in our courses. Despite the fact that many of these students suffer from math anxiety and will admit to hating mathematics, when we give them space to explore mathematics and bring their existing knowledge to the problem, they can make some amazing mathematical discoveries. Inquiry-based learning (IBL) is perfect structure to provide these type of opportunities. In this paper, we will examine one inquiry-based investigation in which MLA students were given the space to look for patterns which resulted in some original discoveries.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript describes an inquiry-based learning (IBL) activity in a Mathematics for Liberal Arts (MLA) course in which students explore the Rascal Triangle, identify patterns in the array, and produce what the authors present as original mathematical discoveries.

Significance. If the reported student patterns are verifiably novel, the work would supply concrete classroom evidence that IBL can enable non-STEM students to generate authentic combinatorial insights; the paper is otherwise a descriptive account of observed student activity rather than a derivation or theorem.

major comments (1)
  1. [Abstract and body] Abstract and body (description of student discoveries): the repeated assertion that the patterns constitute 'original discoveries' is load-bearing for the central claim, yet the manuscript contains no literature comparison or citation check against prior work on the Rascal Triangle (or equivalent combinatorial arrays such as Pascal's triangle variants or Riordan arrays). Without this verification it is impossible to determine whether the identities are new contributions or rediscoveries.
minor comments (2)
  1. The manuscript would benefit from explicit separation of verbatim student statements from author commentary on the patterns.
  2. A brief statement of how the Rascal Triangle was introduced to the students (initial array, prompting questions) would improve reproducibility of the activity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the importance of verifying the status of the reported patterns. We address the major comment below and outline revisions to improve the manuscript's clarity and context.

read point-by-point responses

  1. Referee: [Abstract and body] Abstract and body (description of student discoveries): the repeated assertion that the patterns constitute 'original discoveries' is load-bearing for the central claim, yet the manuscript contains no literature comparison or citation check against prior work on the Rascal Triangle (or equivalent combinatorial arrays such as Pascal's triangle variants or Riordan arrays). Without this verification it is impossible to determine whether the identities are new contributions or rediscoveries.

    Authors: We agree that the manuscript lacks an explicit literature comparison, which limits the ability to assess absolute novelty. Our primary focus is the educational process: demonstrating that IBL in an MLA course can lead non-STEM students to independently derive combinatorial identities through pattern-seeking. The phrase 'original discoveries' was meant to describe findings generated by the students themselves, not necessarily new to the mathematical literature. To strengthen the paper, we will (1) add a brief review of existing work on the Rascal Triangle (including its relation to Pascal's triangle and Riordan arrays), (2) clarify the scope of our claims to emphasize student agency rather than priority, and (3) note any overlaps or distinctions found. This revision will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

No derivation chain or load-bearing steps present; descriptive educational report

full rationale

The manuscript is an observational report on an inquiry-based learning exercise in which MLA students explore the Rascal Triangle and note patterns. It advances no mathematical derivations, predictions, fitted parameters, or first-principles claims. The abstract and provided text contain no equations, no self-citations used to justify uniqueness, and no ansatzes or renamings of known results. The central claim concerns student capability under IBL and requires no reduction to inputs by construction. This is a self-contained descriptive paper with no circularity risk under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a descriptive educational report rather than a formal mathematical derivation, so the ledger contains no free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5609 in / 973 out tokens · 28998 ms · 2026-05-24T19:56:21.030719+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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