Productive Failure

A Successful “Productive Failure” Lesson

There are multiple methods for accomplishing this “boost” in conceptual understanding through struggle (problem-solving followed by instruction ${}^1$, comparing with contrasting cases ${}^9$, impasse-driven learning ${}^{12}$, preparation for future learning ${}^{13}$, inventing to prepare for learning ${}^{14}$); however, research suggests the most effective methodology is “productive failure” ${}^{2,3,4,6,10}$.

A successful “Productive Failure” Lesson has the following features:

• An initial exploration phase that activates your students’ prior knowledge
• A task that requires student collaboration to generate possible solutions using partners or small groups
• A “critique and refine” phase that prompts students to question whether their solutions would work and explain why or why not
• A direct instruction phase that builds on the student-generated solutions

This last criterion probably poses the biggest challenge to teachers, as it requires adapting direct instruction to the specific ideas raised during student explorations. Utilizing the specific ideas students have generated is essential in fostering the sense of agency and increased intrinsic interest that accompanies productive failure. When students see their ideas successfully built upon to get to the correct solution, it helps them frame their failures as steps toward success. It then becomes easier to integrate the correct solution with the problem model they began building during the critique and refine phase.

An example of “Productive Failure” from the Research

Manu Kapur, the researcher behind the concept of productive failure, assigned four 9th-grade classrooms to learn about the formula for variance over the course of four, 55-minute class periods. The classrooms were split such that two classrooms received four class periods of direct instruction and two classrooms received two class periods of direct instruction following two productive failure lessons.

In the first group receiving only direct instruction, the teacher started with the formula for standard deviation and modeled how to use it in the context of a data analysis problem. These students then practiced using the formula on other data sets in small groups and were given other practice problems to take home as homework.

The other two classrooms spent the first two class periods in productive failure by generating their own formulas for variance. They were given the distribution of “goals scored” by 3 soccer players, then worked in small groups to create a quantitative index to show which player was the most consistent. No instruction or scaffolding was provided for the first two class periods. In the final two periods, direct instruction with practice was done as with the other group. Because they did not start direct instruction until the 3rd class period, they did not complete as many practice problems as the direct instruction group, and they were not given homework problems.

The productive failure classrooms group “failed” to generate the correct formula for standard deviation; however, they did stumble upon some pretty sophisticated insights into the concepts underlying standard deviation, including central tendencies, qualitative representations of score distribution, frequency methods, and deviation methods. Despite completing fewer practice problems than the direct instruction (DI) group, the productive failure group scored:

• No differently than the DI group on procedural fluency (calculating standard deviation for a given dataset)
• Significantly higher than the DI group on data analysis questions (comparing Means and SDs between two samples)
• Significantly higher than the DI group on conceptual insight questions (e.g. dealing with outliers)
• Significantly higher than the DI group on learning transfer (applying their learning to a new situation that was not explicitly taught, in this case, normalization)

The students who were given a chance to grapple with the concepts underlying the formula for standard deviation and generate their own solutions, even if none of those solutions were “correct,” were better prepared to receive the instruction as more than just a rote formula to memorize and repeat. They were able to understand the meaning behind the formula, the “why,” and flexibly apply it in ways that were not explicitly taught.

Allowing students to be their own meaning-makers, even if some precious class time must be sacrificed, pays dividends to their deep understanding of the learning goal.

When Not to use a “Productive Failure” Approach

There are some circumstances in which a lesson designed around productive failure won’t work as intended. The biggest exception is when students don’t have enough prior knowledge to do the initial solution generation phase. There’s no point in ‘activating prior knowledge’ if there’s none to activate! In cases like this, it may be useful to revisit the “productive failure” lesson later in the unit, when students can build on skills that they have already practiced.

Another circumstance in which productive failure might not be the best choice is when the “cognitive load” demands are too high for your students. A simple way to decide if the “cognitive load” is too high is to ask yourself, “How many things do my students have to actively and effortfully keep in mind in order to solve this?” If the answer is more than 3, the cognitive load is too high for a “productive failure” lesson ${}^1$. How much a student needs to actively process in working memory will decrease as skills become automated. Meaning, the more experience students have with a requisite skill, the more capable they will be of benefitting from a “productive failure” lesson. It may be necessary to build skills utilizing explicit instruction before implementing productive failure.

How to Use “Productive Failure” Within the OpenStax Algebra 1 Curriculum

An example of how to use “productive failure” in Algebra 1 can be seen in Lesson 2.9: Solutions to Inequalities. The lesson opens with a warm up that asks students to write several solutions to the inequalities $y\le 9.2$ and $7(3-x)>4$ and explain their answers. This activates prior knowledge of equations from Lesson 1.4 and inequalities from throughout Unit 2. This continues the process of “sensemaking” around what a “solution” means. Importantly, this warm up does not require a correct answer. It is a space for ungraded exploration where it’s okay to “fail.”

After completing the warm up activity, students should break into pairs or small groups for student collaboration to match a series of number lines to inequalities. The goal of this activity progression is to encourage students to notice what the difference is between the open circles and the closed circles, and whether the direction of the arrow matters.

Next, encourage students to refine their thinking. In this example, they would do that by comparing the different number lines. What makes them similar? What makes them different? Listen to student responses and remember them, so you can build upon them for the next step. Pose pointed questions, such as, “Could ‘3’ be a solution to this inequality?” This is the “critique and refine phase” of the productive failure lesson. Students must either integrate their answer into the mental model they are building or adjust the model.

In preparation for the “direct instruction phase” of the lesson, one of the key features you need to listen for is the difference between the open and closed circles. This determines if the two quantities can be equal. You’ll want to remember the student generated reasoning that you overheard in Lesson 2.9.1, so you can bring it up in Lesson 2.9.5 and build on the students’ responses. Here, students are trying to think about the difference between an equality and inequality. In your direct instruction, emphasize the boundary conditions of the inequality, referencing the reasoning you overheard in Lesson 2.9.1. If it’s allowed to equal 2, we need to decide if there are other variables or other values that will also make that relationship true. If there are multiple values, that means it’s an inequality. This builds upon the observations the students are making during the number line activity.

In conclusion

There won’t always be a graceful way to incorporate all of the elements of “productive failure” into every lesson, and that’s okay. There is still utility in letting your students explore, experiment, discover, and build their own mental models before you explain the “right” way to do things. Even if you can only bring in some elements, like making comparisons, generating solutions, or building on prior student reasoning, you are still getting closer to the ideal outcome of improving learning transfer, conceptual understanding, and intrinsic motivation. At the very least, incorporating productive failure communicates to your students that you trust them to be “sensemakers,” that you see them as “mathematicians,” and that being “wrong” is just a step on the way to getting it “right.” These are worthy goals in and of themselves, even without a boost in learning transfer.