A
5×5 taxonomy of paper crafts, used to exemplify the Expanded Taxonomy of
Thinking
Srijan Sanchar
· Working draft v0.1 · 24
October 2026
|
Status: provisional. The paper-craft grid is a proposed morphological
coordinate system, not an established taxonomy. The pairing with the Taxonomy
of Thinking was made by judgement and has not yet been tested by independent
raters. Fit ratings are the author's assessment, not evidence of learning
outcomes. |
•
Builds a 5×5 taxonomy of paper crafts (origami,
kirigami and paper cutting, paper engineering, flexagons, playing and learning
cards) from two independent axes: where a sheet's meaning sits, and how much
authorship the maker exercises.
•
Pairs every cell of the Taxonomy of Thinking
(all 25) with one paper-craft activity that enacts that kind of thinking.
•
Gives an exercise per cell that can be used to
strengthen that kind of thinking, and rates how well the craft carries it.
•
Reports what the pairing suggests about the taxonomy
itself, as hypotheses to test, and where paper falls short.
All 25 cells of
the Taxonomy of Thinking received a craft exemplar, and each of the 25 craft
cells was used exactly once. Fit is rated on a three-point scale:
•
Strong (11 cells): the craft itself forces the
thinking move; it is hard to do the activity without making the move.
•
Moderate (12 cells): the craft invites the move,
but a facilitator prompt is needed to bring it out.
•
Stretch (2 cells): the craft is the occasion;
the thinking is carried mainly by discussion.
The pattern by row is the main finding. Paper
carries thinking well from the Perceptual row through the Integrative row. The
Reflective / Meta row is the weakest: one strong, two moderate and two stretch
pairings. For that row the craft supplies something to reflect on, and a
debrief protocol supplies the reflection.
Because one
author chose every pairing, the one-to-one fit is a design convenience, not
evidence that the two grids share a structure. Section 8 proposes a blind
pairing test that would give that claim real weight.
The reference
is the Expanded Taxonomy of Thinking published on srijansanchar.com on 11
November 2025. It crosses a Structural Axis (depth and organisation of
cognition, from perception to meta-awareness) with a Directional Axis
(orientation or purpose, from discovering to transforming). Cell names in this
document are as published; the codes T1.1 to T5.5 (row.column) are added for
reference. The full grid, with the paired craft cell shown in each cell,
appears in Section 5.
Scope.
Sheet-based crafts in which paper is the main structural material: origami,
kirigami and paper cutting, paper engineering (pop-ups and mechanisms),
flexagons and kaleidocycles, and playing and learning cards. All of these are
toys or artefacts in the sense of the request.
•
P1 · Surface. Meaning is on the sheet: marks,
faces, diagrams.
•
P2 · Fold. Meaning is in the creases.
•
P3 · Cut and Join. Meaning is in what is removed
and how pieces are joined.
•
P4 · Move. Meaning is in motion: mechanisms.
•
P5 · System. Meaning is in many sheets under
rules.
•
A1 · Replicate. Follow a given pattern; the
outcome is fixed.
•
A2 · Vary. Change a parameter of a given
pattern; the outcome shifts predictably.
•
A3 · Play. Interact with a pattern or rule set;
the outcome emerges in use.
•
A4 · Compose. Combine units into a new whole.
•
A5 · Invent. Create a new form, mechanism or
rule set.
•
They are independent. Each of the 25 pairs
corresponds to a real practice, so no cell had to be forced.
•
They differ on purpose from the thinking axes.
If the craft axes copied Perceptual-to-Reflective and Discover-to-Evolve, the
pairing would be a tautology. With different axes, the pairing is a test.
•
Their ordering is a judgement. Surface-to-System
and Replicate-to-Invent are plausible orders, not measured scales.
Originality. Teaching mathematics,
design and game thinking through paper is long established. Flexagons date from
1939 and were popularised by Martin Gardner in 1956; origami mathematics and
pop-up engineering have their own literatures; card-based learning is already
surveyed in the Card-Based Gamification discussion piece. What is offered here
is the specific two-axis grid and its pairing with the Taxonomy of Thinking.
Any A5 · Invent outcome should pass a prior-art check before it is called new.
|
Form Principle ↓ / Authorship → |
A1 · Replicate |
A2 · Vary |
A3 · Play |
A4 · Compose |
A5 · Invent |
|
P1 · Surface Meaning is on the sheet: marks, faces, diagrams. |
P1·A1 Copy a pattern Colour or copy a Ganjifa card face, kolam or Sanjhi motif
from a model. |
P1·A2 Re-skin Keep a deck's structure; change what suits and ranks mean. |
P1·A3 Sort and match Memory, snap, attribute-sorting and odds-drawing on single
faces. |
P1·A4 Spread and tableau Lay cards in meaningful layouts: tableau, tarot spread,
storyboard row. |
P1·A5 Design a new card Create a card face or deck logic that breaks a default
(shape, count, suit rule). |
|
P2 · Fold Meaning is in the creases. |
P2·A1 Fold from diagram Fold a crane or boat from a printed diagram. |
P2·A2 Trim and tune Adjust a fold angle, flap or weight on a paper plane or
kite. |
P2·A3 Fold-play Free folding and crumpling; fortune-teller; fold-and-fly
tests. |
P2·A4 Fold a tessellation Repeat a fold rule to build a form: dragon-curve strip,
Miura-ori, waterbomb. |
P2·A5 Design a crease pattern Build a new model from crease-pattern theory. |
|
P3 · Cut and Join Meaning is in what is removed and how pieces are joined. |
P3·A1 Cut from template Cut a stencil or standard snowflake from a given template. |
P3·A2 Vary the symmetry Same cut, different fold count: snowflakes of order 4, 6,
8. |
P3·A3 Fold-and-cut prediction Predict the shape revealed by unfolding a single cut. |
P3·A4 Assemble nets and strips Join nets into polyhedra; quilled mosaics; group-built
structures. |
P3·A5 Design a pop-out structure Origamic architecture: a new cut-and-fold piece from one
sheet. |
|
P4 · Move Meaning is in motion: mechanisms. |
P4·A1 Build a standard mechanism V-fold pop-up card or hexaflexagon from a template. |
P4·A2 Tune a mechanism Shift a fold or tab; observe what opens, closes or jams. |
P4·A3 Flex and explore Flip a flexagon to find hidden faces; spin a kaleidocycle. |
P4·A4 Compose mechanisms Multi-layer pop-up scene or paper automaton telling a
story. |
P4·A5 Invent a mechanism Prototype a new transformable form through paper mock-ups. |
|
P5 · System Meaning is in many sheets under rules. |
P5·A1 Follow a standard set Learn a card game from printed rules; assemble a standard
Sonobe cube. |
P5·A2 Vary the system House rules; change deck composition or unit count; run
what-ifs. |
P5·A3 Play a card game Trick-taking, bluffing, set-collecting: outcomes emerge
from rules and hidden information. |
P5·A4 Compose a deck or build Deck-building under constraints; kusudama and modular
sculptures. |
P5·A5 Invent a game or unit system Design new rules, a new deck or a new modular unit. |
Each cell shows the published
thinking-cell name and, after the arrow, the craft cell paired with it. Shading
shows fit: green strong, amber moderate, grey stretch.
|
Structural Level ↓ / Directional Mode → |
Exploratory (Discover) |
Diagnostic (Understand) |
Evaluative (Judge) |
Constructive (Create) |
Transformative (Evolve) |
|
1 · Perceptual (Embodied / Sensory) |
T1.1
Observe / Sense / Enact → P2·A3 Fold-play (Strong) |
T1.2
Detect / Notice Anomalies → P2·A1 Fold from diagram (Strong) |
T1.3
Sense-Judge / Feel Value → P3·A2 Vary the symmetry (Moderate) |
T1.4
Compose / Arrange → P1·A4 Spread and tableau (Moderate) |
T1.5
Attune / Harmonize → P2·A2 Trim and tune (Moderate) |
|
2 · Associative (Link-Based / Divergent) |
T2.1
Connect / Analogize → P1·A2 Re-skin (Strong) |
T2.2
Trace / Contextualize → P1·A1 Copy a pattern (Moderate) |
T2.3
Compare / Empathize → P5·A3 Play a card game (Strong) |
T2.4
Combine / Story-Form → P4·A4 Compose mechanisms (Strong) |
T2.5
Reframe / Redefine → P4·A3 Flex and explore (Strong) |
|
3 · Analytical (Rule-Based / Logical) |
T3.1
Hypothesize / Model → P3·A3 Fold-and-cut prediction (Strong) |
T3.2
Deduce / Infer / Analyze → P4·A1 Build a standard mechanism (Moderate) |
T3.3
Evaluate / Estimate / Classify → P1·A3 Sort and match (Moderate) |
T3.4
Design / Algorithmize / Formalize → P2·A4 Fold a tessellation (Strong) |
T3.5
Reconstruct / Theorize → P2·A5 Design a crease pattern (Moderate) |
|
4 · Integrative (Systemic / Strategic) |
T4.1
Explore Systems / Simulate Scenarios → P5·A2 Vary the system (Moderate) |
T4.2
Diagnose Systems / Contextual Mapping → P4·A2 Tune a mechanism (Strong) |
T4.3
Balance / Strategize / Prioritize → P5·A4 Compose a deck or build (Strong) |
T4.4
Architect / Co-Create / Organize → P3·A4 Assemble nets and strips (Moderate) |
T4.5
Integrate / Synthesize / Reconcile → P3·A5 Design a pop-out structure (Moderate) |
|
5 · Reflective / Meta (Self-Referential / Ethical) |
T5.1
Inquire / Question Assumptions → P1·A5 Design a new card (Moderate) |
T5.2
Self-Assess / Adapt / Iterate → P4·A5 Invent a mechanism (Moderate) |
T5.3
Clarify Values / Moral Imagine → P5·A1 Follow a standard set (Stretch) |
T5.4
Redesign Thinking / Reclassify → P5·A5 Invent a game or unit system (Strong) |
T5.5
Evolve Consciousness / Re-Envision Reality → P3·A1 Cut from template (Stretch) |
For each
Taxonomy of Thinking cell: the paired craft cell, how the craft enacts the
thinking move, an exercise that can be used to strengthen that kind of
thinking, and the fit rating. Exercises are suggestions for facilitators and
have not been piloted.
|
Thinking cell |
Paper-craft cell |
How the craft enacts the thinking |
Exercise to strengthen it |
Fit |
|
T1.1 Observe / Sense / Enact Exploratory (Discover) |
P2·A3 Fold-play |
Paper shows its grain, stiffness and crease memory only
through the hands. Free folding is perception before any plan exists. |
Give each person an unmarked sheet and three minutes with
no goal. Then unfold it and describe aloud what the creases say about how the
paper behaved. |
Strong |
|
T1.2 Detect / Notice Anomalies Diagnostic (Understand) |
P2·A1 Fold from diagram |
A wrong crease early on shows up later as a model that will
not close. The paper itself signals the anomaly. |
Hand over a diagram with one deliberately wrong step.
Participants locate the faulty step from the misfit alone, before checking
the diagram. |
Strong |
|
T1.3 Sense-Judge / Feel Value Evaluative (Judge) |
P3·A2 Vary the symmetry |
Snowflakes cut after 4, 6 or 8 folds look different at a
glance. The judgement 'this one is better' arrives before any reason does. |
Cut three versions from one template and rank them in ten
seconds. Then write the reasons and compare them with the first reaction. |
Moderate |
|
T1.4 Compose / Arrange Constructive (Create) |
P1·A4 Spread and tableau |
Placing cards in a row, grid or spiral shapes sensory
elements into an order that carries meaning before anyone explains it. |
Give 12 picture cards. Arrange them to say something
without words, let a partner read the arrangement, and compare the two
readings. |
Moderate |
|
T1.5 Attune / Harmonize Transformative (Evolve) |
P2·A2 Trim and tune |
A paper plane or kite flies well only when the maker
adjusts to the day's air. The craft is a loop between hand, paper and wind. |
Fly one plane indoors and outdoors. Change one thing per
flight and record what the air asked for. |
Moderate |
|
Thinking cell |
Paper-craft cell |
How the craft enacts the thinking |
Exercise to strengthen it |
Fit |
|
T2.1 Connect / Analogize Exploratory (Discover) |
P1·A2 Re-skin |
Keeping a deck's structure (suits, ranks, court cards)
while changing what they mean forces a cross-domain mapping. It parallels the
deck grammar proposed in the Card-Based Gamification discussion piece (Oct
2026). |
Map suits and ranks of a standard deck onto a subject the
group knows well. Then find where the mapping breaks and note what the break
teaches. |
Strong |
|
T2.2 Trace / Contextualize Diagnostic (Understand) |
P1·A1 Copy a pattern |
Copying a heritage motif (a Ganjifa face, a Sanjhi design,
a kolam) slowly enough invites the question of what each element means and
where it came from. |
Copy one card or motif, then trace three links: the
community that makes it, the story it draws on, and how it is used. |
Moderate |
|
T2.3 Compare / Empathize Evaluative (Judge) |
P5·A3 Play a card game |
Trick-taking and bluffing games force each player to model
what another player can see and wants. Hidden hands make perspective-taking
unavoidable. |
Pause a game mid-hand. Each player writes what the player
on their left holds and why. Reveal, then discuss the gap between belief and
fact. |
Strong |
|
T2.4 Combine / Story-Form Constructive (Create) |
P4·A4 Compose mechanisms |
A pop-up scene joins separate mechanisms (a rising door, a
turning wheel) into one sequence that opens as a story. |
Plan a three-page pop-up story in which each reveal is a
plot turn. Storyboard the mechanisms before any cutting. |
Strong |
|
T2.5 Reframe / Redefine Transformative (Evolve) |
P4·A3 Flex and explore |
Flexing a hexaflexagon turns the same strip into a face
that was hidden all along. The object does not change; the view does. |
Give each person a flexagon with a problem statement on one
face. They restate the problem on each new face from the position it reveals. |
Strong |
|
Thinking cell |
Paper-craft cell |
How the craft enacts the thinking |
Exercise to strengthen it |
Fit |
|
T3.1 Hypothesize / Model Exploratory (Discover) |
P3·A3 Fold-and-cut prediction |
Before cutting a folded sheet, the maker predicts the
unfolded result. That is a testable hypothesis resolved in seconds. |
Predict, sketch, cut, unfold, and score the prediction.
Vary the number of folds to form a rule of thumb, then test the rule on a new
case. |
Strong |
|
T3.2 Deduce / Infer / Analyze Diagnostic (Understand) |
P4·A1 Build a standard mechanism |
After building a V-fold pop-up from a template, the maker
can deduce why it rises: the fold geometry forces it to. |
Build the card, then change one dimension in a mock-up.
Predict from the geometry alone whether it still closes flat, then test. |
Moderate |
|
T3.3 Evaluate / Estimate / Classify Evaluative (Judge) |
P1·A3 Sort and match |
Sorting cards by attribute is classification. Drawing from
a deck gives a live feel for odds and probabilistic estimation. |
Estimate the chance of a given hand before drawing. Run 20
draws, compare outcome with estimate, and discuss the gap. |
Moderate |
|
T3.4 Design / Algorithmize / Formalize Constructive (Create) |
P2·A4 Fold a tessellation |
Folding a strip in half repeatedly and unfolding it reveals
a pattern generated by a rule (the dragon curve). A fold rule is an algorithm
executed by hand. |
Write the fold rule as steps, predict the crease sequence
for the next iteration, then fold to check the prediction. |
Strong |
|
T3.5 Reconstruct / Theorize Transformative (Evolve) |
P2·A5 Design a crease pattern |
Origami design theory (rules of flat-foldability, folding
axioms) is a rebuilt conceptual structure. Designing from the theory, not by
imitation, is theorising. |
Unfold a known model and record its crease pattern. State
which rules it obeys, alter the pattern while keeping those rules, and test
whether it still folds flat. |
Moderate |
|
Thinking cell |
Paper-craft cell |
How the craft enacts the thinking |
Exercise to strengthen it |
Fit |
|
T4.1 Explore Systems / Simulate Scenarios Exploratory (Discover) |
P5·A2 Vary the system |
Changing deck composition or house rules and replaying the
game is a cheap simulation of how a system behaves when a parameter moves. |
Run a short game three times, changing one rule each time.
Log how strategy, length and fairness shift, and name the scenario that
surprised the group. |
Moderate |
|
T4.2 Diagnose Systems / Contextual
Mapping Diagnostic (Understand) |
P4·A2 Tune a mechanism |
In a pop-up, moving one fold a few millimetres decides
whether the whole structure opens, jams or tears. It is a lived leverage
point and constraint. |
Move one fold in 2 mm steps across five trials. Mark where
behaviour changes and map which fold is the leverage point. |
Strong |
|
T4.3 Balance / Strategize / Prioritize Evaluative (Judge) |
P5·A4 Compose a deck or build |
Building a deck from a limited pool forces trade-offs among
strength, cost and variety. |
Give a card pool and a deck-size limit. Each team builds a
deck, plays another team, and explains what it gave up. |
Strong |
|
T4.4 Architect / Co-Create / Organize Constructive (Create) |
P3·A4 Assemble nets and strips |
A group assembling polyhedra from nets or a quilled mosaic
must divide the work, agree interfaces and keep tolerances compatible. |
Split a structure among four builders who cannot see one
another's parts until joining. Debrief on interface failures. |
Moderate |
|
T4.5 Integrate / Synthesize / Reconcile Transformative (Evolve) |
P3·A5 Design a pop-out structure |
Cutting breaks continuity and folding preserves it.
Origamic architecture reconciles the two opposed operations in one sheet that
stands up. |
Decide which cuts and which folds a target form needs,
build it, then list where cut and fold conflicted and how the design resolved
it. |
Moderate |
|
Thinking cell |
Paper-craft cell |
How the craft enacts the thinking |
Exercise to strengthen it |
Fit |
|
T5.1 Inquire / Question Assumptions Exploratory (Discover) |
P1·A5 Design a new card |
Traditional Ganjifa decks are round and commonly use more
suits than the Western deck. That shows rectangular cards, four suits and 52
cards are choices, not necessities. |
List five assumptions an ordinary deck makes, break one,
then design and play-test the result. Record what the broken assumption had
been quietly doing. |
Moderate |
|
T5.2 Self-Assess / Adapt / Iterate Diagnostic (Understand) |
P4·A5 Invent a mechanism |
Paper prototypes are cheap enough to fail often, so the
maker can watch their own design process in the pile of failed versions. |
Keep every prototype. Label each with what changed and what
was learned, then sort the pile by type of mistake. |
Moderate |
|
T5.3 Clarify Values / Moral Imagine Evaluative (Judge) |
P5·A1 Follow a standard set |
The rules of a standard game embed values: what is
rewarded, who is protected, what counts as cheating. Learning them from the
printed rule sheet makes those values visible. Here the paper is the
occasion; the ethical work happens in the discussion. |
After one round, list what the rules reward and what they
punish. Propose a rule change that expresses a different value and say who
would gain or lose. |
Stretch |
|
T5.4 Redesign Thinking / Reclassify Constructive (Create) |
P5·A5 Invent a game or unit system |
Designing new rules or a new modular unit is designing a
thinking structure: it fixes categories, permitted moves and what counts as
winning. |
Design a deck that teaches a subject using the deck
grammar, play-test it, and reclassify the content wherever play exposes
categories that do not fit. |
Strong |
|
T5.5 Evolve Consciousness / Re-Envision
Reality Transformative (Evolve) |
P3·A1 Cut from template |
In stencil-cut traditions such as Sanjhi the empty space
carries the image. Repeated careful cutting can be a contemplative discipline
in which attention, not originality, is the work. This is a replicate-level
activity paired with a transformative cell, so it holds only if treated as
sustained practice. |
Cut the same stencil slowly several times across a week.
After each cut, note what attention caught that the previous one missed. |
Stretch |
|
Structural level |
Strong |
Moderate |
Stretch |
|
1 · Perceptual (Embodied / Sensory) |
2 |
3 |
0 |
|
2 · Associative (Link-Based / Divergent) |
4 |
1 |
0 |
|
3 · Analytical (Rule-Based / Logical) |
2 |
3 |
0 |
|
4 · Integrative (Systemic / Strategic) |
2 |
3 |
0 |
|
5 · Reflective / Meta (Self-Referential / Ethical) |
1 |
2 |
2 |
Strong fits appear in every row, but the
Reflective row has only one, and both stretch pairings sit there.
|
Thinking row |
P1 Surface |
P2 Fold |
P3 Cut |
P4 Move |
P5 System |
|
1 · Perceptual |
1 |
3 |
1 |
0 |
0 |
|
2 · Associative |
2 |
0 |
0 |
2 |
1 |
|
3 · Analytical |
1 |
2 |
1 |
1 |
0 |
|
4 · Integrative |
0 |
0 |
2 |
1 |
2 |
|
5 · Reflective / Meta |
1 |
0 |
1 |
1 |
2 |
Fold activities cluster in the Perceptual and
Analytical rows; Move activities in the Associative row; System activities in
the Integrative and Reflective rows. A plausible reading is that single-sheet
forms serve perception and rule-based thought, while many-sheet systems serve
integration and reflection. Because the author made the pairings, this pattern
may be an artefact of those choices and should be treated as a hypothesis.
•
Add a reversibility or commitment note per cell.
Paper operations differ in how reversible they are: a flexagon flips back
freely, a fold leaves a crease memory, a cut cannot be undone. Thinking moves
differ similarly, with exploratory moves cheap to reverse and constructive or
transformative moves often committing. A reversibility note might help design
exercises. This is untested.
•
Embodiment is a channel, not only a row. Several
analytical cells were best enacted with the hands (predicting a cut, folding a
dragon curve). The Perceptual row marks a depth of structure, but an enactment
channel appears usable at every depth. This is an observation about teaching,
not a proposed change to the structure.
•
Projection may be a shared structure. A crease
pattern is a flat drawing from which a folded form is reconstructed; each face
of a flexagon is a view of one object's state space. This resembles the idea of
a lower-dimensional view of a higher-dimensional object that underlies PAC and
Visual RISE. It is a parallel worth testing, not a claim that they are the
same.
•
The craft grid is missing a Deconstruct mode.
Unfolding a finished model to read its crease pattern served two thinking cells
(T1.2, T3.5) yet has no craft cell of its own. A v0.2 of the craft taxonomy
could add reverse-engineering to the Authorship Gradient.
The two stretch
pairings (T5.3, T5.5) and several moderate ones in the Reflective row show that
paper supplies an occasion for reflection, not the reflection itself. For those
cells the exercise should carry a written debrief with fixed prompts, such as:
what did the rules or the repetition reward, what did you assume without
noticing, and what would you change.
•
Blind pairing test. Give three to five raters
the 25 craft cells and the 25 thinking cells without the pairing and ask for
their best matches. Agreement with the pairing here, and among raters, would
show whether the fit is structural or an artefact of one author's choices.
Several cells have credible alternatives, so some disagreement is expected.
•
Pilot a few cells. Run five or six exercises
(for example T1.2, T2.5, T3.1, T4.2, T4.3, T5.4) with a small group, and record
which thinking cell participants report using, to compare with the cell
intended.
•
No outcome evidence yet. Nothing here shows that
these exercises improve thinking. The evidence base for card and paper
activities in learning is mixed, as the Card-Based Gamification discussion
piece itself notes.
•
Possible home for the grid. The craft cell could
serve as the 'activity' field of the design canvas proposed for the Card Lab in
that discussion piece, with the thinking cell as its 'learning goal'.
•
Craft taxonomy v0.2. Add a Deconstruct mode,
test the row order, and run the prior-art check for any Invent-level outcome
that is to be called new.
•
Expanded taxonomy of thinking, srijansanchar.com/Blogs
(11 November 2025): the 5×5 grid and cell names reproduced in Section 5.2.
•
Card-Based Gamification of Learning,
srijansanchar.com/Blogs (17 October 2026): deck grammar, Card Lab and design
canvas referred to in Sections 6 and 8.
Background statements about origami
flat-foldability rules, fold-and-cut, the dragon curve, flexagon history,
Ganjifa decks and Sanjhi stencil-cutting come from general knowledge, were not
retrieved for this document, and should be checked before any external use.