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Advanced Practice Questions

shuffle the deck

Twenty problems of the same kind in a row teach you to do that kind, not to recognise it. In real life, and in exams, nobody tells you which method a problem needs. So once each method works on its own, practise from a shuffled set with no labels, and let every question start with the hard part: deciding what kind it is.

  • Challenging to start
  • Life-Changing impact
  • Learn it in 30 days
  • Feels worse, works better
three decks, one shuffled set
Three kinds of problem learned in blocks, then dealt as one unlabelled set, with an error log that routes each miss At the top, three decks practised one at a time: Speed, distance ÷ time; Percentages, change ÷ original; Ratios, total ÷ parts. Below, a Sunday set of 6 problems dealt from all three, with no labels, so no two of the same kind sit together. Beside each, the kind the solver named in pencil before solving, and the answer. Problem 1: 45 km in 1 h 30 min. Average speed? Named speed. 30 km/h, right. Problem 2: A £40 jacket is cut to £30. Percent off? Named % change. Wrote 33%, should be 25%: wrong method. Problem 3: Share £60 in the ratio 2 : 3. Named ratio. £24 and £36, right. Problem 4: Rent goes from £800 to £840. Percent rise? Named % change. Wrote 4%, should be 5%: slip. Problem 5: Blue to white, 3 : 1. Blue in two litres? Named ratio. 1.5 litres, right. Problem 6: 30 km at 60 km/h, 30 more at 30. Average speed? Named speed. Wrote 45 km/h, should be 40 km/h: didn't know. 3 of 6 right. The error log routes each miss: wrong method means more of that kind in the next mix; a slip means checking the arithmetic; didn't know sends the speed problems back to their own deck for one short block. first, one kind at a time Speed distance ÷ time a block of eight Percentages change ÷ original a block of eight Ratios total ÷ parts a block of eight Sunday · shuffled, no labels 3 of 6 1 45 km in 1 h 30 min. Average speed? speed 30 km/h 2 A £40 jacket is cut to £30. Percent off? % change 33% → 25% ✗ wrong method 3 Share £60 in the ratio 2 : 3. ratio £24 and £36 4 Rent goes from £800 to £840. Percent rise? % change 4% → 5% ✗ slip 5 Blue to white, 3 : 1. Blue in two litres? ratio 1.5 litres 6 30 km at 60 km/h, 30 more at 30. Average speed? speed 45 km/h → 40 km/h ✗ didn't know error log · where each miss goes wrong method more percentages in next week's mix slip check the sum before moving on didn't know back to the speed deck, one block

Each kind turns up twice and never twice running. Three misses, three different kinds of miss, and each goes somewhere different: only the one that didn't know goes back to its deck.

Once you can do each kind alone

Try it now.

Two or more kinds of problem that look alike: types of maths problem, verb tenses, chess openings, diagnoses, painters' styles.

  1. Learn each method on its own first

    Something brand new is easier in a block: a worked example, then a few of the same kind until it works.

    Mixing too early just makes everything feel impossible.

  2. Shuffle them into one set, no labels

    Take problems from every kind, strip off the headings, and deal them so no two of the same kind sit together.

  3. Name the kind before you solve it

    Write which method it needs in the margin first. That choice is exactly what blocked practice never trains.

  4. Log every miss by kind

    Wrong method, slip, or didn't know. They look the same on a score sheet and need different fixes.

  5. Send each miss where it belongs

    Wrong method: more of that kind in the next mix. Slip: a checking habit. Didn't know: back to a short block, then into the mix again.

Expect it to feel worse. Your practice scores will drop when you start mixing. That isn't a sign it's failing; in the study below, the people who mixed scored lower in practice and far higher a week later.

Build a mixed set

Shuffle your own.

Name two to four kinds of problem and put a few of each underneath, one per line. The page deals them into one set with the labels off and no two of a kind together. Mark each one as you go, and it tells you where each miss belongs.

    Mixed set

    not dealt yet

      Your shuffled set appears here, with the labels off.

        Nothing you type is saved or sent anywhere. Write the set out, or deal a fresh one next session.

        What the study found

        Worse in practice, three times better a week later.

        In 2007 Doug Rohrer and Kelli Taylor taught college students to find the volumes of four unusual solids, in two practice sessions a week apart. Half practised in blocks: a tutorial on one solid, then its problems, then the next. The other half read all four tutorials and did the same problems shuffled.

        During practice the blockers looked far better, 89% right against 60%. A week after the second session, everyone sat a test of new problems. The mixers scored 63%; the blockers, 20%.

        The honest limits: one kind of maths, a small group of college students, a lab, and a test one week on. Mixing helps when problems look alike and the hard part is choosing the method. Shuffling unrelated things, like geometry with French verbs, gives you nothing to tell apart, so there's no reason to expect the same gain.

        Practice scores against test scores a week later, for blocked and mixed practice During practice, blocked practice scored 89% and mixed practice 60%. On the test a week later, blocked scored 20% and mixed 63%. During practice both sessions Blocked 89% Mixed 60% A week later new problems, no help Blocked 20% Mixed 63%
        Rohrer & Taylor (2007), Experiment 2: percentage of problems right, averaged over both practice sessions, and on the test one week after the second.

        The idea

        Why it works.

        In a block, the method is chosen for you by the chapter heading, so you only ever practise carrying it out. A mixed set makes you notice what tells one kind of problem from another, which is the part that matters when a problem turns up with no heading at all.

        It also spaces things out for free: by the time a kind comes round again, you have to pull it back from memory instead of copying what you did a minute ago. Both make practice harder, and both are why it lasts.

        When to mix, and when not

        Habits that help.

        • Block first, then mix. A method you can't do at all needs a short block before it joins the set.
        • Mix look-alikes. The gain comes from telling similar things apart, so put confusable kinds together.
        • Keep old kinds in. Each set should carry a few from earlier weeks, or you'll lose them quietly.
        • Judge it a week later. Not by how practice felt. Practice scores flatter blocks.

        Go deeper

        Keep going.

        Practise this in HourLife.

        In Learning: Resurface to remember. Track the books, courses, podcasts, videos and articles you are working through, log where you are, and keep what you take from them.

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