How to study math, and why rereading the textbook fails

To study math, stop rereading and start redoing. Read one worked example closely, close the book, and redo the same problem from a blank page. Then do a sibling problem with new numbers. Then do a mixed set where you have to choose the method. Keep an error log that records the exact line that went wrong, not "I messed up". Math is a performance skill, and watching it performed is not practising it.

Why rereading the textbook (and watching videos) fails for math

Here is the pattern. You read the chapter and every step makes sense. You watch a video, the person on screen does the problem, and you nod along. Then the test puts a problem in front of you and nothing comes. You understood it in class and you cannot do it now.

Nothing is wrong with you. Two different skills are involved, and you trained the wrong one. Understanding a step when you read it is recognition: the line is on the page and your job is to agree with it. Doing the problem on a test is recall: the line is not on the page, you have to produce it, and first you have to decide which method the problem wants. Rereading trains recognition. It also produces a feeling of fluency, because the material is familiar, and that feeling gets mistaken for knowing. In the largest review of study techniques, rereading was rated low utility (Dunlosky et al. 2013). It takes time and does not move the mark.

Videos are the same trap in a different format. A video you only watch is a textbook you only read. The person doing the problem is practising. You are watching.

The four moves that do move marks are in the hub for how to study: answer first, space it, mix it, sleep. This page is the math version of the first three. You learn a procedure by producing it, not by recognizing it.

Step 1: One worked example, read like a mechanic

Worked examples are the right way to begin a new procedure. Students who studied worked examples learned algebra procedures faster and made fewer errors than students given the same problems to solve unaided (Sweller & Cooper 1985). The reason is load: a new procedure takes all of your working memory, and solving it cold leaves nothing over for learning the pattern (Sweller 1988). The example carries the load for you, once.

But read it like a mechanic, not like a reader. For every line, say two things: what changed from the line above, and why. Then cover the next line with your hand and predict it before you look. If your prediction matches, move on. If it does not, that line is the one you did not understand, and you found it before the test did.

Not ten examples. One, read this closely, takes about five minutes. Ten examples skimmed teach less than one example read line by line.

Step 2: Redo it cold, then a sibling

Close the book. Take a blank page. Redo the same problem from memory, every line. Then open the book and mark your work line by line against the example. Not "roughly right": line by line. A missing step counts as a miss.

This is retrieval practice for procedures. The example showed you the steps; the redo asks you to produce them. The gap between the two is exactly what you need to learn, and you cannot see it any other way.

Then the sibling: the same problem type with new numbers. The textbook's exercises are siblings of its examples, usually in order. Do one, cold, and mark it. If the redo and the sibling are both clean, the procedure is yours for now. If not, read the example once more the mechanic's way and redo it tomorrow.

Step 3: Fade the steps

There is a smoother path from example to problem than jumping. Faded worked examples remove the steps one at a time from the end: the next example has its last line missing and you supply it; the one after has its last two lines missing; and so on, until the example is a bare problem (Renkl & Atkinson 2003). Each step you fill in is retrieval, and the load rises gradually instead of all at once.

You can do this yourself with any textbook. Take a worked example and cover it from the bottom up. First cover only the last line and write it. Then cover the last two lines. Then three. By the time you are covering the whole solution you are solving the problem, and you got there in steps you could manage.

Step 4: The error log that names the line

Most students "go over their mistakes" by looking at the right answer and nodding. That is rereading again. The useful move is to find the exact line where your work left the correct path, and to write down the rule that line broke. Errors teach when they come with corrective feedback that says what was wrong and why (Metcalfe 2017). A red X is not feedback. The line is.

Keep a four-column log. Copy this table into a notebook.

Date	Problem	The exact wrong line (copied as you wrote it)	The rule it broke, in your words
Nov 6	Textbook 3.4 #7	dy/dx = 4(3x^2 + 1)^3	The chain rule needs the inner derivative. I forgot to multiply by 6x.

The third column is the hard one. You have to find the line, and when you cannot see it yourself, a teacher, a friend who got it right, or a step-by-step answer compared against your own work will show you. When a step-by-step answer teaches and when it replaces the thinking has the rule for using one. Chapter 4 of the free playbook is the error log in full.

Then the redo rule: every logged problem is redone cold two days later, from the problem alone. If it is clean, date it. If it is not, log it again. A problem is not learned until you have done it right on a later day without looking. How to study for a math test is mostly this: the log, redone.

Step 5: Mix the problem types

Homework is usually blocked: twenty problems of one type, the type the section just taught. Blocked practice teaches the recipe. It does not teach the one thing the test adds, which is deciding which recipe this problem wants, with no section heading to tell you.

Interleaving fixes that. When students practised mixed problem types instead of one type at a time, their test scores were higher, even though the practice itself felt harder and went worse (Rohrer & Taylor 2007). In classroom studies with middle-school students, interleaved practice over several months produced higher scores on a later test than blocked practice did (Rohrer, Dedrick & Stershic 2015). It feels worse and tests better. Interleaving explained has the full case.

To shuffle a textbook: write the problem numbers from three sections on slips of paper, mix them in a cup, and draw. Do each problem without looking at which section it came from. Your first job on every problem is to name the method before you touch the numbers.

Worked example: the chain rule, done the five-step way

The chain rule is the eighth idea on the MCV4U path, and the one most students lose marks on first. Here are the five steps on it. The rest of the course is on MCV4U's course map.

Step 1, the example, read like a mechanic

Differentiate y = (3x^2 + 1)^4.

1. The outer function is u^4, where u = 3x^2 + 1. Its derivative is 4u^3. (What changed: the power came down in front and dropped by one. Why: the power rule, applied to the outside.)
2. The inner function is 3x^2 + 1. Its derivative is 6x. (The power rule again, on the inside; the constant 1 disappears.)
3. Multiply: dy/dx = 4(3x^2 + 1)^3 · 6x. (Why: the chain rule says outer derivative times inner derivative.)
4. Tidy: dy/dx = 24x(3x^2 + 1)^3. (4 × 6 = 24, and the x moves to the front.)

Step 2, the redo and the sibling

Close the page and redo it from a blank sheet. Then the sibling: differentiate y = (2x^3 − 5)^3. Outer: 3(2x^3 − 5)^2. Inner: 6x^2. Product: 3(2x^3 − 5)^2 · 6x^2 = 18x^2(2x^3 − 5)^2. Check the number: 3 × 6 = 18.

Step 3, the fade

The next sibling comes with its last line blank: y = (x^2 + 4)^5. Outer: 5(x^2 + 4)^4. Inner: 2x. Product: you write it. (It is 10x(x^2 + 4)^4, since 5 × 2 = 10.)

Step 4, the log

The common wrong line is "dy/dx = 4(3x^2 + 1)^3" with nothing after it. The rule it broke: the inner derivative was not multiplied in. Log it in those words. Redo it in two days.

Step 5, the mixed set

Five derivatives, five different rules, in no order. Cover the right two columns and name the method before you differentiate.

Differentiate	Method	Answer
x^5	power rule	5x^4
x^2 sin x	product rule	2x sin x + x^2 cos x
x / (x + 1)	quotient rule	1 / (x + 1)^2
(2x + 3)^3	chain rule	6(2x + 3)^2
e^(3x)	exponential, with the chain rule	3e^(3x)

Check the quotient: the top is (1)(x + 1) − (x)(1) = x + 1 − x = 1, over (x + 1)^2. Check the chain: 3(2x + 3)^2 × 2 = 6(2x + 3)^2.

The week before a math test

Compress the five steps into a week. One table, one row a day.

Day	What to do
7 to 3	One or two ideas a day, all five steps on each. Each idea comes back once later in the week as a cold redo.
2	A full mixed set under time: every method of the unit, shuffled, marked, every error logged.
1	Redo the whole error log cold. Stop early. Sleep.

The general day-by-day is in the day-by-day test plan, and the grid that spreads the ideas across the week is in the weekly grid. If the week does not hold, drop to one idea a day and keep the mixed set. The mixed set is the part that resembles the test.

When a solver or an answer helps, and when it replaces the thinking

A step-by-step answer is a worked example. Used after your own attempt, it is useful: read it the mechanic's way, find the line where your work diverged, log it, redo it cold. Used before your attempt, it is rereading in disguise, and it feels like progress for the same reason rereading does. There is a second problem with answer apps. In our own reading of their reviews, students describe trusting an answer that turned out to be wrong, and a student who cannot yet do the problem cannot tell which answers those are. The check is the same either way: attempt first, compare second, and anything you cannot verify yourself is not an answer yet.

Summary

1. Read one worked example like a mechanic: for every line, what changed and why; cover the next line and predict it.
2. Redo it cold from a blank page and mark it line by line; then do a sibling with new numbers.
3. Fade the steps: cover the solution from the bottom up, one more line each time.
4. Keep the four-column error log, and redo every logged problem cold two days later.
5. Mix the problem types before the test, and name the method before you touch the numbers.