Trying Math Problems Before Instruction: Is It Worthwhile?

Productive failure is a planned sequence. Students first attempt a complex problem, then receive teaching that organizes and corrects their ideas^[1]. The word “failure” can sound harsher than the activity really is. You are allowed to produce an incomplete or wrong solution.

A review that pooled 53 studies and 166 comparisons found that problem solving before instruction worked better overall than instruction before problem solving^[2]. The advantage was larger when activities closely followed productive failure principles^[2].

That finding supports trying first. It does not support struggling indefinitely without help. The useful unit is the attempt followed by instruction, not the attempt alone^[2].

In one seventh-grade mathematics study, students tried complex average-speed problems before receiving a teacher-led explanation^[1]. Their initial solutions were unsuccessful. Even so, they later did better than directly taught students on both familiar and complex test problems^[1].

The same basic pattern has appeared with university students. In one online physics experiment with 78 undergraduates, students either explored a problem before teaching or practised it afterward^[3]. The explore-first students were less accurate during the activity but showed better understanding on the later assessment^[3].

These findings make an important distinction. Poor performance during a first attempt does not automatically mean the time was wasted. The attempt may prepare you to notice what the later explanation resolves.

Trying first is not the same as being abandoned with a difficult question.

One college biochemistry study compared worked examples, productive failure, and two kinds of guided investigation^[4]. Worked examples, productive failure, and guided investigation all led to better results on similar new problems than investigation without helpful structure^[4]. The approaches did not differ in basic knowledge^[4].

Although that study was not about mathematics, it warns against aimless exploration. A first attempt needs a clear problem, relevant prior knowledge, and instruction afterward.

University mathematics examples offer cautious support. In a linear algebra course, four course groups using preparatory problems earned better final exam results than earlier groups^[5]. However, participation was voluntary, and there was no separate comparison group studying at the same time^[5]. That means the course results cannot prove that the preparatory problems caused the improvement.

A matrix theory course used a similar weekly structure. Students received homework before the ideas were formally discussed^[6]. In a survey, students said this increased their workload but also improved their understanding^[6]. That tells us how they experienced the course, not whether the method alone raised their marks.

Imagine that next week’s set asks you to differentiate

$$y=x^x.$$

You have used the power rule on expressions such as $x^5$. You have also differentiated exponentials such as $2^x$. However, $x^x$ does not fit either familiar form exactly.

Before opening a worked example, write what makes the problem unusual:

• In $x^5$, the exponent is constant.
• In $2^x$, the base is constant.
• In $x^x$, both the base and exponent change.

You might first try the power rule and write $x x^{x-1}$. Mark that step with a question mark. The ordinary power rule assumes a constant exponent, so its condition is missing here.

Next, look for another representation. Because $a^b=e^{b\ln a}$, you can rewrite

$$x^x=e^{x\ln x}$$

for $x>0$. Even if you cannot finish, you have identified a possible route. You also have a precise question for the lecture: how do logarithms help when both the base and exponent vary?

Now study the formal method. Let $y=x^x$. Take the natural logarithm of both sides:

$$\ln y=x\ln x.$$

Differentiate both sides:

$$\frac{y'}{y}=\ln x+1.$$

Finally, multiply by $y=x^x$:

$$y'=x^x(\ln x+1).$$

Your wrong power-rule attempt now has a purpose. It reveals exactly why the familiar rule does not apply. The worked solution supplies a route that handles both changing parts.

One plausible reason this sequence helps is that the first attempt gives the explanation a specific job. Instead of receiving several unfamiliar steps at once, you can compare each step with a route you already considered.

Here is one workable routine adapted from attempt-then-instruction designs^[1][2].

1. Choose one preview problem. Pick a question connected to ideas you already know. Do not preview the entire assignment.
2. State the goal in words. Write what the problem asks you to find, prove, or explain.
3. List relevant knowledge. Record definitions, formulas, graphs, or earlier problem types that might connect.
4. Try more than one representation. Rewrite an expression, draw a graph, make a table, or test a simple case.
5. Mark uncertainty clearly. Use a question mark beside any step whose rule you cannot justify.
6. Stop after a short, focused effort. The aim is to identify routes and gaps, not to win a contest against the problem.
7. Study the formal explanation. Compare its first important choice with your own first choice.
8. Redo the problem without notes. Then solve a similar problem to check whether you can choose the method yourself.

Your notes from the attempt matter. “I got stuck” gives you little to compare. “I tried the power rule, but the exponent was not constant” identifies the exact gap.

None of the university studies here tested first-year calculus^[4][3][5][6]. The university studies covered biochemistry^[4], physics^[3], linear algebra^[5], and matrix theory^[6].

The result can also change with the learners and topic. A review found that instruction first tended to work better for children in grades two through five and for broad problem-solving skills^[2]. In a separate experiment with 122 younger children, teaching the main idea before difficult equation problems produced better learning than delaying that teaching^[7].

The approach does not win in every subject. Two studies with tenth-grade students found no advantage over direct instruction when the topic was social science research methods^[8]. These results caution against treating productive failure as a universal rule.

Nor is working in a group clearly required. One experiment found no difference in learning between individual and collaborative first attempts^[9].

Before one upcoming lecture, open a single unfamiliar problem. Write the goal, try a plausible method, and mark the first step you cannot justify. Then attend the lecture or study the worked example with that gap in view.

Afterward, close the explanation and solve the problem again. Your target is not a correct first attempt. It is a sharper second attempt after the teaching arrives.