Easy Math Practice Is Not the Same Thing as Learning

The reason it felt easy is structural, not a failure of effort. When researchers counted 13,505 practice problems in six widely used mathematics textbooks, fewer than one in ten were mixed in among problems of a different kind — about one or two a school day^[1]. Everything else sat in a block under its own lesson, so you can assume the method carries over instead of picking a strategy from the problem itself^[1].

That picking is the step the page removed^[1]. A test question has no heading above it.

A review of decades of motor-skill and memory research concluded that how well you do while practicing is often a misleading sign of how much will still be there later, and that people mistake smooth practice for learning^[2]. The careful word is "often". Speed does not always lie; it is just not the measurement you want.

One experiment shows both halves at once, and it is not about mathematics. People studied paintings by several artists, each person seeing some artists grouped together and others shuffled among the rest. Shown a painting they had not studied, they named its painter better when that artist had been shuffled. Yet asked afterwards which arrangement helped more, most picked grouping, after their own scores had shown otherwise^[3]. What failed was their judgment, not their effort.

Here is what blocking hides.

$$x^2 - 5x + 6 = 0, \qquad 2x^2 - 3x - 2 = 0, \qquad x^2 - 4x + 1 = 0, \qquad x^2 - 4 = 0$$

The first is the friendly kind. Two numbers multiplying to $6$ and adding to $-5$ are $-2$ and $-3$, so $(x-2)(x-3) = 0$ and $x = 2$ or $x = 3$.

The second has a leading coefficient, so the bracket hunt changes: $(2x+1)(x-2) = 2x^2 - 3x - 2$, giving $x = -\frac{1}{2}$ or $x = 2$.

The third does not factor over the integers: the only integer pairs multiplying to $1$ are $1, 1$ and $-1, -1$, adding to $2$ and $-2$, never $-4$. So you reach for completing the square or the quadratic formula:

$$x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$$

Substituting back confirms it: $(2+\sqrt{3})^2 - 4(2+\sqrt{3}) + 1 = 7 + 4\sqrt{3} - 8 - 4\sqrt{3} + 1 = 0$.

The fourth is a difference of squares: $(x-2)(x+2) = 0$, so $x = \pm 2$, and spotting that skips the bracket hunt.

None of that arithmetic is hard. The hard part is the first fifteen seconds, deciding which situation you are in. Twenty problems of the first kind never ask.

Calculus has the same shape. Differentiating $x^3$, $x^3 + \frac{1}{x}$, $x^2 e^x$ and $\sin(x^2)$ gives $3x^2$, $3x^2 - \frac{1}{x^2}$, $(2x + x^2)e^x$ and $2x\cos(x^2)$ — power rule, a rewrite before the power rule, product rule, chain rule. Under a heading that says "The Product Rule", you never notice that only one is a product.

Across 54 ordinary seventh-grade mathematics classes, assignments ran two ways for four months: one mixed the problem types, the other blocked them. Both groups finished with the same mixed review assignment^[4]. A month later the seventh-graders sat a test nobody had warned them about, and the mixed group averaged 61% against 38%^[4]. It was registered in advance, ran in normal classrooms, and needed no teacher training^[4].

Two experiments with college students doing mathematics point the same way. In the first, they learned one kind of problem, and spreading practice across several sessions beat one sitting on a test a week later^[5]. In the second, other college students learned several problem types, and mixing the types beat taking one type at a time on a test a week later^[5].

Researchers use the name "desirable difficulties" for study conditions that feel like hard work but leave you knowing more^[6]. They also name the trap: because these methods feel effortful and feel like failing, learners avoid them or give them up early^[6].

So expect this from your first mixed set. It will probably take longer, and it will feel like you are getting more wrong. You will feel sure last week's way was better — the feeling the painting study recorded in people whose own scores had shown it false^[3].

Five changes, none needing new materials.

1. Break the runs. Reorder your set so no two consecutive problems require the same strategy — the idea the seventh-grade trial built its assignments on^[4].
2. Reach back. Pull some problems — a third is a rule of thumb, not a tested dose — from chapters you finished weeks ago. Spreading practice across sessions beat one sitting for those college students^[5].
3. Hide the headings. Copy the questions onto blank paper without section titles, so nothing announces the method before you read^[1].
4. Mix the confusable ones. Factoring, completing the square and the quadratic formula all aim at quadratics; only the last two always work. A review that pooled many studies found mixing helped most when the things learned were easy to confuse^[7].
5. Write one yourself. Combine two methods from different weeks into a single question. You cannot write it without deciding what separates them.

A rough trigger, this one opinion rather than finding: if three problems in a row go by without a pause to decide, the set has stopped asking anything.

When we build a problem set at Learn4Less, the ordering is part of the work: you should not be able to tell which method comes next.

That same review concluded mixing can genuinely help, but that what you learn matters^[7]. The reviewers warned against carrying it everywhere: for memorizing words and for explanatory texts, mixing can leave you worse off than grouping^[7]. Do not carry a result about shuffling mathematics problems to a vocabulary list.

One more limit, again opinion. The first few times you meet a method, some repetition is reasonable — you cannot choose between methods you cannot yet execute. Start mixing once the steps are no longer the hard part.

None of this was measured on the subject most readers here study. The classroom trial was seventh-graders on seventh-grade material^[4]; the mathematics experiments used college students on introductory problem types^[5]. No study followed a high-school or calculus class, so applying this to derivatives is an extension, not a tested finding.

Every mixed assignment in these studies was arranged by a researcher or teacher, and none tested when to start mixing^[1][4][5]. Nobody has checked a student reshuffling their own set at the kitchen table, which is what this post asks.

The longest gap between practice and test here is one month^[4], so what mixing does for a June final covering February material is unknown. Two sources need narrow reading: the review of mixed practice is a caution as much as a recommendation, so none of its overall numbers appear here^[7]; and the "desirable difficulties" paper organizes others' evidence rather than reporting its own study^[6].

What survives is still worth acting on. Two groups of seventh-graders practiced over the same four months. Only the order differed. A month later it was 61% against 38%^[4].