Focused and diffuse thinking: why trying harder can keep you stuck¶

Level: 101 · for anyone who has stared at one problem for an hour and got nowhere

One line: Focused thinking takes small steps from where your first idea landed, so it climbs the nearest hill and stops at its top, however long you keep at it; stepping back takes big steps, which find the right hill but not its top. You need both, one at a time: focus, step away, focus again.

The picture¶

Barbara Oakley and Terrence Sejnowski's Learning How to Learn, written with Alistair McConville for children and teenagers (2018), gives its chapter 2 the title "Easy Does It: Why Trying Too Hard Can Sometimes Be Part of the Problem". It opens with a photo from 2004: thirteen-year-old Magnus Carlsen has got up from the board in the middle of a rapid game and wandered off to watch other games, while Garry Kasparov, then the best player in the world, sits thinking. The game was drawn. (Kasparov won their second game, and with it the match; the book tells only the first.) The book's lesson: sometimes we need to lose concentration so we can think more clearly.

The book says the brain works in two ways. In focused mode you pay attention to one thing: a math problem, a teacher, a puzzle. You need it first, to switch on the parts of the brain the problem needs. In diffuse mode the mind is relaxed and wanders: daydreaming, doodling, walking, showering. That is where new connections between ideas often come from. Learning goes back and forth between the two, and you can be in only one at a time.

The picture that explains it is a pinball table. In focused mode the rubber bumpers are close together, and the ball bounces around one small area, along trails laid by thoughts you have had before. In diffuse mode the bumpers are far apart, and the ball crosses the whole table, touching few bumpers on the way. Oakley used the same picture for adults in A Mind for Numbers (2014).

A pinball table is a search, and a search can be run. This page runs it.

The table¶

The program makes a problem out of 60 positions in a row, each with a score, and the answer is the position with the highest score. Focused mode steps to a neighbour, one position left or right, whenever the neighbour scores higher: bumpers next to each other. Diffuse mode looks at every seventh position across the whole table: bumpers far apart. The first idea lands at position 10.

Verified output of focused_and_diffuse.py — regenerated by tools/run_examples.py, never hand-typed.

1. The problem: 60 positions, and the score of each

    90 |                                ###
    80 |                       #       #####      ##        ###
    70 | ###                  ####    ######     ####      ######  #
    60 |#####       ##       ######  ########   ######     #########
    50 |######     ####     ###########################   ##########
    40 |#######   ######    ########################################
    30 |################# ##########################################
       +------------------------------------------------------------
                  S                      *

   S is where the first idea lands (position 10, score 45).
   * is the best answer (position 33, score 97).
   There are several hills; only one is the highest.

2. Focused mode only: small steps uphill from S

   step  position  score
      0        10     45
      1        11     54
      2        12     60

   After 2 steps both neighbours are lower, so the climb stops
   at score 60: the top of the nearest hill, not of the highest.

   Trying harder, with the same small steps:

   steps allowed   final position   final score
               3               12            60
              10               12            60
             100               12            60
            1000               12            60

   A thousand steps end where three did. Effort is not the problem;
   the step size is. Every step away from the peak goes downhill, so
   a focused climber never takes it.

3. Diffuse mode only: one look at every 7th position

   position     0   7  14  21  28  35  42  49  56
   score       63  34  54  66  58  80  80  48  70

   The best look is position 35, score 80. It has found the
   right hill, far from S, but not its top: 80, not 97.
   Wide bounces cross the table and miss the details.

4. Both, one at a time: focus, step back, focus again

   phase                          positions visited   best score
   focused from S                                 3           60
   diffuse survey                                 9           80
   focused from the best look                     3           97
   total                                         15

    90 |                                ###
    80 |                       #       #####      ##        ###
    70 | ###                  ####    ######     ####      ######  #
    60 |#####       ##       ######  ########   ######     #########
    50 |######     ####     ###########################   ##########
    40 |#######   ######    ########################################
    30 |################# ##########################################
       +------------------------------------------------------------
        d      d  S F d      d      d    * d      d      d      d

   S start, F where focus got stuck, d the diffuse looks, * the answer.
   15 looks in all reach score 97, which a thousand focused steps
   did not. The survey picks the hill; focus climbs it.

5. Every starting point, focused mode only

   stuck at position   score   starting positions   how many
                  33      97                28-38         11
                  53      87                48-57         10
                  23      80                18-27         10
                  42      80                39-42          4
                  43      80                43-47          5
                   2      78                  0-7          8
                  59      74                58-59          2
                  12      60                 8-12          5
                  13      60                13-17          5

   Focus alone finds the answer from 11 of 60 starts (18%): only when
   the first idea already landed on the right hill. Where you start
   decides where focus ends.

Trying harder does not move you¶

Section 2 is the chapter's title in numbers. Focused mode climbs from S in two steps to a score of 60 and stops, because both neighbours are lower. Then the program lets it try harder, with 10, 100 and 1000 steps, and it ends at 60 every time. Being stuck here has nothing to do with effort. It comes from the size of the step: to reach the higher hill you first have to go down, and a focused climber never takes a step down.

This has a name in psychology: the Einstellung effect (German for "attitude" or "set"). In Abraham Luchins's water-jar experiments (1942), people who had solved several problems with one long method went on using it on problems that had a far shorter solution, and many failed a problem the long method could not solve, one that people who had not been primed solved easily. The first idea laid a trail, and the ball kept rolling down it. Oakley's books use the word, and the program shows the mechanism: a trail is a hill, and focus climbs whatever hill it starts on.

Stepping back finds the hill, not the top¶

Section 3 is diffuse mode alone. Nine wide looks across the table find the best region, around position 35, far from anywhere focus would go. But the best look scores 80, not 97: bumpers seven apart skip over the peak. A daydream can suggest where the answer is, and it does not work out the details.

Section 4 puts them in the order the book gives: focus first, step back, focus again. Fifteen positions in all reach 97, which a thousand focused steps did not. The survey picks the hill; focus climbs it. And section 5 shows how much depended on luck when focus worked alone: it finds the answer from only 11 of the 60 starting points, the ones that already landed on the right hill.

What the model leaves out¶

The program shows one half of the book's claim, the half about step size. It does not show the other half: that the diffuse mode works on what focus has loaded. Here the survey could run without any focused work first. In a head it cannot; a walk after an hour on a problem is different from a walk with no problem in mind, and the book insists the focused session comes first. Nor does the model say anything about brains. "Focused" and "diffuse" are the book's teaching words, not scientific terms; the nearest ideas in neuroscience are the brain's task-positive networks and its default mode network, active in rest and mind-wandering, and the match is loose.

What experiments do support is the effect itself. Psychologists call it incubation: setting a problem aside for a while and coming back to it. A review of 117 studies (Sio and Ormerod, 2009) found a real, moderate benefit, larger for problems with many possible answers, and larger when the break came after a long period of work on the problem, as the book says it should. The classic story is from mathematics. Henri Poincaré wrote that after weeks of failing on a problem about what he called Fuchsian functions, he set it aside for a geology trip, and the key idea came to him as he put his foot on the step of a bus, while talking about something else. He then checked it at his desk, at leisure: focus again.

What the book says to do¶

Work first, then break. A break is useful only when there is a problem loaded. The book's tool for this is the Pomodoro: 25 minutes of focus with the phone away, then a short reward that is truly different, such as a walk, music or a snack. That rhythm is focus, diffuse, focus, with a timer.
When stuck, change the step size, not the effort. If ten minutes more on the same approach has changed nothing, section 2 says a hundred will not either.
On a test, hard start, then jump to easy. The book's advice for exams: look at a hard problem first, and when you get stuck, move to an easy one and come back. The hard one keeps working in the background while you do the easy ones.
Start homework early. Diffuse mode needs time between sessions, and a problem started the night before it is due never gets it. This is also why spaced retrieval spreads study out.
A break is not a phone. Switching to messages or videos is new focused work, which fills the attention the diffuse mode was using.

What else is in the book, and where it is explained here¶

The rest of Learning How to Learn covers ground this library already has a lesson for, told for younger readers with its own pictures:

The book's idea	Here
The octopus of attention: working memory has about four arms	cognitive load
Brain links: practice turns steps into one chunk	cognitive load
Illusions of learning: rereading and highlighting feel like knowing	metacognition
Recall, test yourself, and spread practice out over days	spaced retrieval
Mix problem types, so you learn to choose the method	interleaving
Teach it, explain it simply, ask why	studying vs learning
You can learn what you are not "passionate" about, and get better at it	learned helplessness: start small
Sleep and exercise help learning	the chapter's summary of the evidence
Memory palaces and memorable images; habits against procrastination	not covered here: no program shows them

A longer summary, and how sure each claim is¶

Longer write-ups of this idea, many of them AI-written, go well past the book. A typical one says the model "is not a metaphor", names the brain networks behind each mode, and ends with a daily protocol. Most of its advice is sound and matches the book. The claims under the advice vary a lot in how well they are supported, and weighing them is metacognition applied to advice, as the chapter does for the big four techniques:

Claim	How sure
Setting a stuck problem aside helps, most after a real effort and with an undemanding task in the break	Good evidence: the incubation review above (Sio and Ormerod, 2009). Coming back to the problem is part of the method, not an extra.
Sleep after learning consolidates it, so cutting sleep to study more is a losing trade	Well established. The chapter's summary of the evidence says the same.
Sleep helps insight	Some evidence: in one well-known experiment, more people found a hidden shortcut in a number task after a night's sleep than after the same time awake (Wagner and colleagues, Nature, 2004). There are few studies of this kind.
Curiosity before studying helps memory	Good evidence (Gruber and colleagues, Neuron, 2014), including for unrelated material learned while curious. "It releases dopamine" is the proposed mechanism, not the finding.
Focus and mind-wandering use networks that tend to switch each other off	Broadly supported: the task-control networks and the default mode network are usually anticorrelated, though how strongly depends on how the scans are processed.
The default mode network causes creative ideas	One small study in epilepsy patients with electrodes already in place: stimulating it lowered the originality of ideas (Shofty and colleagues, Brain, 2022). Suggestive, not settled.
Good thinkers switch between the networks at an ideal rate, on an inverted-U	Weak: correlations in brain scans, not a rule you can use to set your breaks.
Diffuse mode is the default mode network, so the model "is not a metaphor"	Overstated. The book's own authors use it as a teaching picture; the match to brain networks is loose, as above.
Short naps of 10–20 minutes trigger hippocampal replay; nature restores attention; TV blocks the diffuse mode	Weak to mixed. Replay is well shown in animals, and a nap that short is mostly light sleep. The evidence on nature is mixed, and the TV claim has no clear source.
"8 hours of study with 4 hours of sleep retains less than 5 with 8"	Invented numbers. The direction fits the sleep research, but no study measured this trade.
Carlsen walks away and "returns with a decisive move"	An embellished story. He does leave the board often, and the game in the book's photo was a draw.

The protocol at the end of such summaries is good advice, and each step already has a reason in this library. Asking yourself a question before you start is count the vowels: know the goal before you read. A focused block and then a real break is this page and McGuire's intense study session. Recalling without notes afterwards is spaced retrieval. Sleep ends the day.

One line in these summaries is worth correcting with the program: "focused mode does not know when it is stuck". In section 2 the climber does know it has stopped, since both neighbours are lower. What it cannot know is whether it stopped on the highest hill: from inside, the top of a small hill and the top of the highest one look the same, because both have only lower neighbours. The signal to step back is therefore the one to use in practice. Stop when you have made no progress, rather than when you feel you are close, because feeling close is exactly what a local peak feels like.

Try it¶

Time your stuck. Next time you are stuck, note the time. After 10 more minutes with no progress, stop, walk for 5, and write down whether the problem looks different when you come back.
Name your first idea. Before starting a problem, write your first approach down. When stuck, ask whether you are still on it: that is the hill you started on.
Change the start. Set START in the program to 30, then to 50. Where does focus end, and does the diffuse survey care?
Change the bumpers. Set FAR to 3, then to 15. Which survey finds the right hill, and how many looks does each cost?

Flashcards¶

The page as a deck of Anki cards: focused_and_diffuse.txt. Import with File → Import. Tags: definition, mechanism, evidence, trap, practice.

Po polsku, w skrócie¶

Barbara Oakley opisuje dwa tryby myślenia. W trybie skupionym (focused) pracujemy uważnie nad jednym zadaniem; w trybie rozproszonym (diffuse) umysł odpoczywa i błądzi: spacer, prysznic, rysowanie bez celu. Obrazem jest flipper: w trybie skupionym odbojniki stoją gęsto i kulka krąży po małym obszarze, po śladach, które już znamy; w rozproszonym stoją rzadko i kulka przelatuje przez cały stół. W danej chwili możemy być tylko w jednym trybie.

Program traktuje to jak szukanie najwyższego punktu na pagórkowatym terenie. Tryb skupiony robi małe kroki pod górę i zatrzymuje się na szczycie najbliższego wzgórza; tysiąc kroków kończy się tam, gdzie trzy, bo do wyższego wzgórza trzeba najpierw zejść w dół. Dlatego „bardziej się starać” często nie pomaga: problemem jest wielkość kroku, a nie wysiłek. Psychologia nazywa to efektem Einstellung. Tryb rozproszony robi duże skoki: znajduje właściwe wzgórze, ale nie jego szczyt. Dopiero oba naraz, jeden po drugim (skupienie, przerwa, znów skupienie), dają najlepszą odpowiedź.

Badania nad inkubacją (odłożeniem problemu na jakiś czas) potwierdzają, że przerwa po długiej pracy pomaga, choć nazwy „tryb skupiony” i „rozproszony” to metafora, a nie termin naukowy. Praktyczne rady z książki: najpierw pracuj, potem rób przerwę (technika Pomodoro: 25 minut skupienia i krótka nagroda); na teście zacznij od trudnego zadania, a gdy utkniesz, przejdź do łatwego i wróć; zaczynaj pracę domową wcześnie. Telefon to nie przerwa, tylko nowe zadanie dla uwagi.

Popularne, dłuższe streszczenia tej idei (często pisane przez AI) twierdzą, że to „nie metafora”, tylko dokładny opis sieci mózgowych. To przesada. Dobrze potwierdzone są korzyści z przerwy po wysiłku, ze snu po nauce i z ciekawości przed nauką. Słabo potwierdzone są twierdzenia o idealnym tempie przełączania się między sieciami i o drzemkach. Liczby w stylu „8 godzin nauki i 4 snu to mniej niż 5 i 8” są wymyślone. Uwaga z programu: wspinacz wie, że stanął, ale nie wie, czy na najwyższym wzgórzu, bo szczyt małego wzgórza wygląda od środka tak samo jak szczyt najwyższego. Dlatego sygnałem do przerwy powinien być brak postępu, a nie uczucie, że rozwiązanie jest blisko.