TL;DR:
Reading a solved example is not the same as being able to solve one. Redo it from a blank page.
Cover the solution, then fade the steps you need to see, one at a time, until you need none.
Keep an error list: the same few mistakes usually repeat.
You read the solved example in your textbook and every line makes sense. You nod your way to the bottom. Then you try the practice problem right below it, and by the second line you are stuck. The example was clear because someone else was doing the thinking.
Maths sticks when you have done the thinking yourself, from nothing, more than once. Here is a routine for turning a solved example into a skill, shown on a quadratic equation.
Why watching the steps is not enough
A worked solution hides the hard part, which is deciding what to do next. Every step is shown after the decision was made. When you follow it, you check each move for sense, which is much easier than producing the move.
So the aim is to move the decisions to you gradually. First you see the whole solution, then you see less and less of it, until a blank page is enough.
What you need
One solved example you have just read, and a similar practice problem.
Paper and a pen, and a sheet to cover the solution.
A page headed "Error list" where you record every mistake along with its cause.
The routine
Read the solution once, asking why: For each line, say why it is allowed, not only what changed.
Cover the solution and redo it from a blank page: Write the problem and solve it with the solution hidden. Uncover only when you are truly stuck, take one hint, and cover again.
Fade the steps: On the next pass, give yourself less: first show only the first step, then only the answer, then nothing.
Change the numbers: Solve a new problem of the same type. If it works, you have the method, not just the example.
Log every error: Write what went wrong and why, in one line.
Worked example: redoing a solved example from a blank page
The example: solve 3x² − 12x + 9 = 0.
The solution you read
Divide both sides by 3: x² − 4x + 3 = 0.
Find two numbers that multiply to 3 and add to −4. They are −1 and −3.
Factor: (x − 1)(x − 3) = 0.
Set each factor to zero: x = 1 or x = 3.
Check by substituting. For x = 1: 3(1) − 12(1) + 9 = 3 − 12 + 9 = 0. For x = 3: 3(9) − 12(3) + 9 = 27 − 36 + 9 = 0. Both work.
The three passes
Pass | What you can see | What you do |
|---|---|---|
1. Cover | Only the problem | Solve from a blank page, peek at one line only if stuck |
2. Fade | Only "divide by the common factor first" | Finish the rest unaided |
3. New numbers | Nothing | Solve 2x² − 10x + 12 = 0 |
Pass three has an answer you can verify. Divide by 2: x² − 5x + 6 = 0. The numbers −2 and −3 multiply to 6 and add to −5, so (x − 2)(x − 3) = 0 and x = 2 or x = 3. Substituting: 2(4) − 10(2) + 12 = 8 − 20 + 12 = 0 and 2(9) − 10(3) + 12 = 18 − 30 + 12 = 0.
What the error list caught
Suppose that on the first blank-page attempt you wrote x² − 4x + 9 after dividing, forgetting to divide the 9 as well. That goes in the list as "divided only some terms by the common factor", with the fix "divide every term". A few lines like that, and you start seeing the same slips in new problems before you make them.
Space the repeats
One blank-page success on Monday tells you the method is in your short-term memory. Do the same example again on Wednesday and once more the following Monday. If Wednesday goes badly, that is useful: it shows the method was borrowed, not owned, and you can fix it while there is time. Each repeat can use new numbers, so you are practising the method rather than recalling one answer.
Mistakes that keep maths from sticking
Copying the solution to "practise". That trains your hand, not your decisions.
Moving on after one success. Try again two days later, with new numbers.
Doing only the problems you can already do. Your error list should point you at the next ones.
Hiding the checking step. Substituting the answer back is cheap and catches many errors.
Treating a wrong answer as the end. Find the exact line where it went off, and write it down.
Your plan for this week
Choose three solved examples from this week's lesson, one per day.
For each, do the cover, fade and new-numbers passes, in that order.
Review the error list on Friday and pick the most common slip to practise.
If you get stuck partway through, DuckyHelper can work the next step out with you on its blackboard rather than hand over the answer. Ask it from any app with one key press; you can try it for 10 minutes with no card.
Frequently asked questions
How many examples should I redo?
Fewer than you think. Three examples done properly beat ten read quickly.
What if I cannot even start from a blank page?
Take the first step from the solution, cover it again and carry on. Next time, take less.
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