Seeing the shape of a problem

In 1987 Jill Larkin and Herbert Simon explained why a diagram is worth ten thousand words, and put a deliberate “sometimes” in the title. A diagram holds no more information than the equivalent text.

What changes is the cost of using it: things needed together sit together, so the eye finds the next step without hunting. Laying a problem out in space is not decoration — it puts related things side by side and makes the gaps visible.

The gaps are the part most people underrate. A list tells you what you have, but rarely what you lack, because every entry looks equally finished. Spread the same material across a table, a wall or a screen and absence acquires a shape: a group with a single item in it, a question nothing points to, a conclusion with no source beneath it.

Relationships you can see

Larkin and Simon’s distinction is simple. In text, information is indexed by position in a list, and each item is adjacent only to the next. In a diagram, information is indexed by location on a plane, so a single element can sit next to many. Modelling problems in physics and geometry, they found three advantages: a diagram groups together what is used together, keeps everything about one element in one place so you need not match labels, and supports inferences perception makes almost for free, such as spotting alternate angles in a geometry figure.

Everyday arrangement uses the same few devices. Proximity says these belong together; a boundary says they form one thing; a line says this bears on that; empty space says nothing is here yet, which a list cannot say at all. Studying how people use space while cooking, packing and assembling, David Kirsh described veteran jigsaw puzzlers sorting edges, corners and sky into separate piles: the coarse grouping narrows the search, so the fine comparison happens within a small pile (Kirsh, 1995).

Take a family choosing where to move. The evidence arrives as a stream — a listing, a school report, a note about the commute, a spreadsheet of costs — and in a message thread each piece is read once and filed by date. Laid out in three groups, one per neighbourhood, the comparison starts to do itself: one group is thick with school information and has nothing on travel; another holds the best house and no school at all. The layout does not make the decision, but it shows which questions nobody has asked.

None of this depends on the claim that the brain processes images 60,000 times faster than text; people who have gone looking for the study behind that number have not found one. The real advantage is more modest: less searching, less to hold in mind, more work handed to perception. Stuart Card, Jock Mackinlay and Ben Shneiderman, adapting Richard Hamming, put it plainly: “the purpose of visualization is insight, not pictures”.

Context that stays put

Card and his co-editors begin their book with long multiplication. Multiplying two two-digit numbers in your head is hard not because the arithmetic is hard, but because the partial results have to be held somewhere until they are needed. On paper they sit outside the mind, and working memory is, in effect, extended. The idea is sometimes called external cognition, and a spatial layout is a larger, more patient version of the same trick.

A layout also keeps its own history. Things stay where you put them, and where you put them was a decision: the cluster pushed to the edge records what you ruled out, the note left between two groups a doubt about where it belonged. Barbara Tversky, in Mind in Motion (2019), names the habit: when thought overflows the mind, the mind puts it back into the world, and organises it there the way it is organised inside. Position is also a strong memory cue in its own right — the old lesson of the method of loci.

Consider a researcher mapping the literature on a question over several months. By summer she cannot recall many of the spring’s papers by title, but she remembers that the small-sample studies sat in the top left, and that two papers with opposite results were placed side by side on purpose. After a fortnight away, the arrangement gives her back much of the argument before she reopens a single paper, because some of those relationships were never written down anywhere else.

Room to try alternatives

Arrangement is cheap to change, which makes it a tool for thinking rather than a record of it. Moving a note from one group to another asks a real question: does this belong here? Kirsh points to Scrabble players shuffling their tiles to prompt new words from the same letters, a habit Moving things to think looks at more closely. Tversky goes further: one of her laws of cognition is that spatial thinking is the foundation of abstract thought.

Comparison is where space earns its keep. A product team weighing three ways to rebuild a sign-up flow can write each option up separately, or set them side by side with the same rows beneath each: cost, risk, what users said, what it rules out later. Side by side, asymmetries are hard to miss — option B has four supportive interviews and no estimate; option C is the only one anyone has costed. Across three documents, the same check means reading each in turn and remembering the others.

Concept mapping turns this into a method. Joseph Novak’s group at Cornell developed concept maps in 1972 to follow how children’s understanding of science changed. A map starts from a focus question; concepts that do not yet fit wait in a “parking lot” rather than being forced in; and the links worth hunting for are cross-links between distant parts of the map, which Novak and Alberto Cañas describe as often representing creative leaps.

From sources to a next step

Research and action usually live in different places. Sources collect in a folder, conclusions go into a document, and the next step lands in a task list, stripped of the reasons behind it. When someone later asks why the team chose option C, the answer has to be reassembled from three places that were never connected.

One view can hold the whole chain: the sources, the findings drawn from them, the decision they support and the step that follows. Lines matter most here. A decision connected to its evidence can be checked; a decision floating on its own can only be trusted. When new evidence arrives, you can see which conclusion it touches.

This is the arrangement a PlansFlow canvas is built around. A project is a single infinite canvas where questions, sources, notes, images, PDFs and generated drafts sit as blocks that can be grouped and connected. A web search answer lands as a block with its sources attached, so a claim arrives already tied to where it came from. When you generate a draft, you choose which blocks inform it, so the grouping you did while thinking becomes the context the model works from. None of this does the thinking for you; it keeps the pieces and their connections in view, which is the precondition for thinking about them together.

When space works against you

The “sometimes” deserves its due. Larkin and Simon were explicit that nothing guarantees an arbitrary diagram is worth ten thousand words: it helps only if it is built to group what is used together, and only for someone who knows how to use it. Card and his co-editors are blunter — depending on the task, a visualisation can make it better or worse. The common failures are predictable.

• Clutter. When everything is near everything, nearness stops meaning anything, and a crowded map can be harder to search than a list, which at least has an order.
• False connections. A line looks like a finding even when it is a guess. Novak and Cañas note that almost any two concepts are related somehow, so links should be chosen selectively and labelled; students who struggle to label a link usually do not yet understand the relationship.
• Sequence. Some material is genuinely linear: a procedure, a recipe, an argument in which each step rests on the last. Text is built for exactly that, and scattering it across a canvas loses the order that carried the meaning.
• Neglect. A layout is a set of claims about how things relate, and claims go stale. Novak and Cañas say plainly that a concept map is never finished; one left untouched for a month records what you used to think.

Even the evidence for mapping comes with conditions. A meta-analysis of 55 studies found that learning with concept maps was associated with better retention, but the size of the benefit ranged from small to large depending on how the maps were used and what they were compared with (Nesbit and Adesope, 2006). Tversky’s first law of cognition fits the case: there are no benefits without costs.

In practice

1. 01

   Group before you connect. Put related items near each other first, and draw a line only when you can write on it, in a few words, what the relationship is.

2. 02

   Leave gaps where they fall. An empty row under one option, or a question nothing points to, is information; resist tidying it away.

3. 03

   Compare alternatives side by side under the same headings, so that a missing estimate or an untested assumption shows up as a blank instead of hiding in a separate document.

4. 04

   Keep each decision in the same view as the sources behind it and the step that follows, so the reasons travel with the task.

5. 05

   Leave procedures and step-by-step arguments as lists, and prune the canvas whenever it stops matching what you now think.