Bad Foundations and Manipulable Objects

Eduardo Ochs

REVIEW 3 major objections 5 minor 2 references

Bad Foundations and Manipulable Objects

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper argues that calculus courses with many poorly-prepared students should treat expressions and trees as basic objects and teach substitution as a visible, manipulable operation before theorems and quantifiers.

desk verdict Honest, clearly-presented Maxima manipulables for weak calculus students; the tools are promising but the learning-transfer claim is untested. read the letter →

arxiv 2602.16026 v2 pith:KQBRO7F6 submitted 2026-02-17 math.HO

Eduardo Ochs This is my paper · ORCID

classification math.HO

keywords badfoundationssubstitutioncalculusteachingMaximamanipulableobjectsexpandableproceduralknowledgecomputeralgebrasystem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is a teaching-strategy proposal for calculus courses that contain many students with what it calls 'extremely bad foundations'—students who can solve x+2=5 by a memorized move but cannot substitute 3 for x to verify it. The author argues that such students cannot benefit from the usual assumption that 'taking a particular case' is obvious, and that the right move is to introduce expressions and trees as basic objects and teach the substitution operation explicitly and early, before theorems and quantifiers. To make substitution visible, the paper develops a set of Maxima macros that display expressions as lisp trees, allow substitution to be annotated as a pretty-printed ':=', and create 'holes' in true expressions so they become exercises. The payoff, if true, is a concrete, low-cost way to help weak students see what mathematicians actually do when they justify a step.

What carries the argument

The key machinery is the substitution operation written as (expr)[var:=value], which the author implements in Maxima as the operator 's'. To handle substitutions of functions, 's' converts function definitions to lambda-expressions on the fly, so that f(g(t))[f(x):=g(g(x)), g(x):=f(f(x))] correctly yields g(g(f(f(t)))). Around this core sits a set of Maxima macros: 'lisptree2' draws the internal Lisp representation of an expression as a two-dimensional tree; 'sss' pretty-prints a substitution in the standard mathematical 'a:=...' notation; 'dpart' and 'substpart' mark sub-expressions, and 'mkholes' replaces chosen sub-expressions by '?' to turn a true expression into an exercise. These toget

What would settle it

A controlled trial with two groups of under-prepared calculus students, one taught with these Maxima manipulable objects early and the other with the standard syllabus, measuring their ability to substitute values into expressions and to justify steps in a chain rule problem after one semester. If the manipulable-object group shows no significant gain over the control group, the central claim is refuted. More directly: if a student who has used the tools still cannot compute d/dx sin(cos(tan(42x))) or cannot explain why (ax+b)[a:=−3,b:=5] = 5−3x, the transfer that the paper hopes for has not o

Watch

Extended reading notes

Core claim

The central claim is that 'when we have lots of students like the ones that I described above it is a good idea to present expressions and trees as some of our most basic objects, and present this substitution operation ... very early in the course, before theorems and quantifiers.' The author offers Maxima-based 'manipulable objects' as a way to make taking a particular case easy to visualize: expression trees drawn from internal Lisp representations, a custom substitution operator 's' that pretty-prints assignments in square brackets, and 'holes' that turn a true equality into an exercise. The discovery is not an empirical result but a designed pedagogical toolset and a specific ordering o

Load-bearing premise

The load-bearing premise is that students with extremely bad foundations who interact with these visual, manipulable objects will develop the pattern-matching and logical skills they lack—the paper explicitly states this has not been rigorously tested, and if it fails, the tools remain just Maxima macros, not a viable teaching strategy.

Editorial extensions

If this is right

If the proposal is right, the calculus course can present substitution as a primary object early on, giving under-prepared students a concrete handle on what 'taking a particular case' means.
The Maxima macros provide a ready way for instructors to turn standard textbook exercises (like chain-rule problems) into manipulable objects with holes, without rewriting the whole course.
Students with extremely bad foundations could see, in a REPL, how changing the substitution changes the result, which may help them shift from memorized procedures to pattern-matching and justification.
The author's 'archetypal case' method for separable ODEs suggests that teaching a single visual example alongside the general method can make the general method believable.
Because the tools run in a free computer algebra system and the non-textbook material is a small part of each class, the approach is feasible even on a crowded syllabus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central claim is true, the same 'substitution as a first-class object' might be tested in other subjects, like algebra or physics problem solving, where students also struggle with plugging in numbers.
The 'holes' mechanism could be extended to automatically generate many variants of an exercise from one template, creating an unlimited supply of practice problems for drill.
The paper's emphasis on visual, manipulable objects suggests a testable prediction: students who interact with the Maxima tools will outperform peers on tasks that require substituting specific values into functions, even on paper exams that do not allow the tools.
The author's choice of Maxima over a proof assistant can be read as a broader point: for this population, immediate visual feedback in a light-weight REPL may be more valuable than formal type-checking, a hypothesis that could be examined by comparing outcomes with a Lean-based intervention.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know about this paper is that it's a thoughtful, honest position/tool paper. Ochs describes specific Maxima functions (lisptreem, lisptree2, the 's' substitution family, holes) that let you visualize expressions as trees and substitutions as operations. The concrete examples are real and I suspect they'd help some students see what 'substituting x=3' means. The author is also unusual in his transparency: Section 7 flatly says the method is 'a work in progress', tested on small classes without rigorous methodology. That honesty is worth taking seriously.

What's actually new is the specific Maxima implementation — the lazy '+' operator, the pretty-printed substitution macros, the hole generator — rather than the underlying ideas. The paper builds on known didactic literature (e.g., Freudenthal, Sfard, Mason) and his own prior work, which he cites. As a piece of engineering, the code sketches look plausible, though I could not verify them without the full definitions, which are not included.

The soft spots are in the pedagogy, not the code. The central claim — that presenting trees and substitution early, before quantifiers, will help students with extremely bad foundations learn to reason — rests entirely on an assumption about learning transfer. The author provides no evidence that interaction with these manipulables produces the 'aha' moment he hopes for. In fact, to understand the tree visualization of f(g(x)), a student like E, who can't substitute x=3 into x+2=5, has to first understand that an expression is a parsed structure with named child positions. That is precisely the competence the tool is meant to build. It's possible the tool creates that understanding through use, but that's the untested hypothesis. So the paper is not a validated teaching strategy; it's a set of promising prototypes.

That said, the paper doesn't overclaim. It is explicitly exploratory. For a reader in math education or CAS-based teaching, the examples are concrete enough to try in a classroom and adapt. For a general research reader, it's a minor piece, but it deserves serious referee time because it's a genuine, clearly-presented proposal that could generate empirical studies. I'd send it to a referee with expertise in both Maxima and calculus pedagogy.

Recommendation: engage with it conditionally — treat it as a source of classroom tools, not as evidence. If the author publishes the code in full and some controlled classroom data, the contribution would be significant.

Referee Report

3 major / 5 minor

Summary. The paper argues that calculus instructors facing many students with 'extremely bad foundations'—students who know mathematics only as memorized procedures—should introduce expressions and trees as foundational objects and teach an explicit substitution operation very early, before theorems and quantifiers (Section 1). The author proposes Maxima as the vehicle: custom operators ('s', 'V', 'sss', 'ssu'), Lisp-tree visualizations ('lisptreem', 'lisptree2'), and 'holes' ('mkholes') turn symbolic expressions, equalities, and justifications into manipulable, expandable objects (Sections 2, 4, 6). The paper includes Maxima input/output examples, a discussion of why Lean is not used (Section 5), and a closing section (7) that explicitly labels the approach a work in progress with only informal classroom trials.

Significance. The paper is genuinely useful as an exposition of a pedagogical idea: it connects a concrete computational tool (Maxima) to documented difficulties with procedural versus conceptual knowledge (Engelbrecht-Bergsten), substitution, and equality (Fischbein, Thomas, Sfard), and it introduces a distinctive visual language for substitutions and 'holes' that could be adapted by other instructors. The Maxima examples are mostly self-consistent and give a clear sense of the intended interaction. The 'holes' idea—rewriting a true expression with unknown parts—is a novel and promising exercise format. However, the paper's central evaluative claim is not supported by evidence; it is a proposal with anecdotal illustrations, not a validated teaching strategy. If the paper is revised to frame its contribution as a testable hypothesis and to provide either pilot data or a detailed evaluation plan, its value would be considerably clearer.

major comments (3)

[Section 1; Section 7] The load-bearing assertion in Section 1—that 'when we have lots of students like the ones that I described above it is a good idea to present expressions and trees as some of our most basic objects'—is stated as a recommendation but is not supported by any empirical evidence. Section 7 explicitly concedes: 'I am treating as a work in progress; I have only a few examples ready, and they were tested only on small classes, and without a rigorous methodology.' There is no control group, no measured learning outcome, no systematic classroom protocol, and no analysis of transfer. The authors should either (a) present this as a conjecture with explicit falsifiable predictions and a plan for evaluation, or (b) include a pilot study, however small, with pre/post measures. In its current form, the abstract and introduction overclaim relative to the evidence.
[Section 6] The paper states at the end of Section 6: 'I haven't shown (yet!) how to transform our expandable proof from section 1 into a manipulable object – I am just showing how to manipulate objects that look like its justifications.' This is a significant missing component. The Section 1 L–M–R proof layout is central to the paper's motivation, but the claimed manipulable-object version of that proof is not provided. The paper would need either to implement and show such a version, or to restrict its claims to the objects that are actually demonstrated (substitution and justifications).
[Sections 4 and 6] A deeper conceptual issue: the target student E cannot substitute x=3 into x+2=5. The proposed visualization relies on the student recognizing that an expression is a parse tree with named child positions (e.g., lisptreem(f(a,b)) shows f applied to two children). For a student who does not already see substitution as a meaningful operation, it is not clear why the tree diagram is more transparent than the original equation. The paper does not address this prerequisite or offer a diagnostic. A concrete test would be to check whether E, after working with lisptree2 and holes, can identify the 'x' in 'x+2=5' as a leaf and can trace how substitution changes that leaf. Without such a bridge, the intervention risks presupposing the very competence it aims to teach.

minor comments (5)

[Section 4.1, p. 10] In the notation example at the bottom of p. 10, '(ax+b) \binom{a:=-3}{b:=5} = -3x+6' contradicts the Maxima output in (%o11), which is 5-3x (i.e., -3x+5). Since b=5, the result should be -3x+5. This error in a central substitution illustration should be corrected.
[Section 4.1] The phrase 'In the notation that we used in section 2' is likely a cross-reference error; the substitution notation was introduced in Section 1 (the chain rule example), not Section 2.
[Front matter] The dedication 'Para Walter Machado Pinheiro, que não leu e não vai ler documento nenhum, e se ler não vai entender' is unprofessional and potentially offensive to a named individual; it should be removed or rewritten.
[Sections 4 and 6] Several custom Maxima functions ('s', 'V', 'sss', 'ssu', 'mkholes', 'lisptreem', 'lazy +.') are used but their definitions are not included in the paper. For reproducibility, provide the code as an appendix or a clear link to a repository/URL. Some references to the author's own materials (OchsNSC2026, OchsMD, etc.) are given, but the main paper should be self-contained at least for the central macros.
[Section 4.1; Section 7] Minor typos: 'obtaing' (Section 4.1) and 'reconstruced' (Section 7) should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the pedagogical proposal is explicitly untested, and the Maxima substitution tools rest on standard lambda-calculus, with self-citations only ancillary.

full rationale

The paper contains no quantitative derivation, no fitted parameters, and no empirical prediction that could reduce to its inputs. Its central claim—that expressions and trees, with an explicit substitution operation, should be introduced early for students with extremely bad foundations—is a pedagogical proposal supported by classroom anecdotes, not by a formal derivation. Section 7 explicitly disclaims rigorous evidence: "I have only a few examples ready, and they were tested only on small classes, and without a rigorous methodology." That is a limitation on evidential support, not a circular step. The substitution machinery is implemented in Maxima by delegating to the built-in `subst` and by translating functions to lambdas; the correctness of the β-reduction order is attributed to the Church-Rosser theorem via [HindleySeldin2008], an external standard result. The paper's self-citations ([OchsNSC2026] for set comprehensions, [OchsEmacsConf2024] for the Maxima interface, [OchsIDARCT] and [OchsMD] for archetypal/general parallelism) are supporting techniques and presentation details; none is load-bearing for the central pedagogical claim. Section 5's "I hope to finish the details for this example soon" is an acknowledged incomplete development, not a disguised circular inference. Terms such as "expandable objects" and "manipulable objects" are new names for familiar ideas, but the paper does not present the renaming itself as a derived result. No step exhibits Eq. X = Eq. Y by construction or a fitted parameter renamed as a prediction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

No free parameters are fitted: the paper makes no empirical predictions. The main load-bearing assumptions are pedagogical (that visualization and drill can repair extremely weak procedural foundations) and technical (that the Maxima code works as described). The invented entities are didactic/software constructs, not new physical or mathematical entities, and only the code pieces have an external falsifiable handle (execution in Maxima).

assumptions (5)

domain assumption Students with 'bad foundations' and 'extremely bad foundations' exist and are characterized by purely procedural knowledge and rejection of trial-and-error.
Defined via classroom anecdotes in Sections 1 and 7; no diagnostic instrument or baseline data, yet the entire target population is assumed.
domain assumption 'Taking a particular case' or substitution is an operation that can be made intuitive through visual/tree representations.
The central method in Sections 1–4 presupposes that visualizing internal structure helps remediate conceptual gaps; this is not tested.
domain assumption Maxima's internal Lisp representation and the custom functions behave as described, including correct beta-reduction behavior.
Sections 4–6 rely on Maxima semantics and Church-Rosser theorem; no formal verification, and several custom functions are only partially defined.
domain assumption Completing 'hole' exercises will transfer into pattern-matching skills and possibly understanding of logic.
Section 7 explicitly hedges with 'it may be possible' and 'they may reach'; an untested learning-transfer hypothesis.
standard math Lambda-calculus substitution with alpha-renaming is well-defined and applicable to school algebra expressions.
Section 4.1 cites Hindley & Seldin and the Church-Rosser theorem; standard background from logic/CS.

invented entities (3)

Expandable objects
purpose: Objects too large to display fully; students interact by toggling parts (game trees, set comprehensions, proofs).
A pedagogical metaphor introduced in Section 2; no falsifiable handle outside the paper.
Manipulable objects with holes
purpose: Expressions/proofs with '?' holes that students modify in Maxima to turn true statements into exercises.
Introduced in Sections 5–6; the learning benefit is asserted, not measured.
Maxima custom operators (s/ss/sss/ssu, V, lisptreem, lisptree2, mkholes, lazy +.) independent evidence
purpose: Visualize and perform substitution, display Lisp trees, and create/pretty-print holes for teaching.
The code snippets in Sections 4–6 specify deterministic behavior; running in Maxima yields checkable outputs, though the full package is not shipped.

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Cite this review

Pith. "Pith review of Bad Foundations and Manipulable Objects." pith.science (2026). https://pith.science/paper/KQBRO7F6

@misc{pith2026260216026,
  author       = {Pith},
  title        = {Pith review of: Bad Foundations and Manipulable Objects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQBRO7F6}},
  note         = {Machine review of arXiv:2602.16026}
}

abstract

Imagine a student -- let's call him `E', and make him a "he" -- that is enrolled in Calculus 2, and who believes that to pass in maths courses he only needs to memorize methods and apply them quickly and without errors. Let's imagine that `E' is an `E'xtreme case of "bad foundations" and that he knows how to solve $x+2=5$ by doing $x=5-2=3$, but he doesn't know how to substitute the $x$ in $x+2=5$ by 3, and the only way that he knows of "testing the solution" is to apply the same method again and check that he got the same result. When we are teaching Calculus to classes that have many students that are extreme cases of bad foundations we need new strategies and tools; for example, we can't pretend that "taking a particular case" is an obvious operation anymore -- instead we need ways to make these operations easy to visualize. This article shows a way to do that using Maxima.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 1 linked inside Pith

[2004]

When Good Teaching Leads to Bad Results: The Disasters of ‘Well-Taught’ Mathemat- ics Courses

Chap. Symbols and Language, pp. 189–223. [SchoenfeldWhenGood] A. H. Schoenfeld. “When Good Teaching Leads to Bad Results: The Disasters of ‘Well-Taught’ Mathemat- ics Courses”. In:Educational Psychologist23.2 (1988), pp. 145–166. [Sfard] A. Sfard.Thinking as Communicating – Human De- velopment, the Growth of Discourses, and Mathema- tizing. Cambridge, 200...

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[2011]

Con- ceptual and Procedural Approaches to Mathematics in the Engineering Curriculum: Student Conceptions and Performance

Chap. Algebra and Technology, pp. 179–202. [EngelbrechtBergsten] J. Engelbrecht, C. Bergsten, and O. K ˚ agesten. “Con- ceptual and Procedural Approaches to Mathematics in the Engineering Curriculum: Student Conceptions and Performance”. In:Journal of Engineering Educa- tion101.1 (2012), pp. 138–162. [FeynmanJoking] R. Feynman.Surely You’re Joking, Mr. Fe...

arXiv 2012

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2004]

When Good Teaching Leads to Bad Results: The Disasters of ‘Well-Taught’ Mathemat- ics Courses

Chap. Symbols and Language, pp. 189–223. [SchoenfeldWhenGood] A. H. Schoenfeld. “When Good Teaching Leads to Bad Results: The Disasters of ‘Well-Taught’ Mathemat- ics Courses”. In:Educational Psychologist23.2 (1988), pp. 145–166. [Sfard] A. Sfard.Thinking as Communicating – Human De- velopment, the Growth of Discourses, and Mathema- tizing. Cambridge, 200...

1988

[2] [2011]

Con- ceptual and Procedural Approaches to Mathematics in the Engineering Curriculum: Student Conceptions and Performance

Chap. Algebra and Technology, pp. 179–202. [EngelbrechtBergsten] J. Engelbrecht, C. Bergsten, and O. K ˚ agesten. “Con- ceptual and Procedural Approaches to Mathematics in the Engineering Curriculum: Student Conceptions and Performance”. In:Journal of Engineering Educa- tion101.1 (2012), pp. 138–162. [FeynmanJoking] R. Feynman.Surely You’re Joking, Mr. Fe...

arXiv 2012