Recently I was asked why it is important for students to experience productive struggle during a math lesson. This is a very fair question, since historically the role of math teacher has been to explain math concepts so clearly that students can’t help but understand.
Why would we ever want to intentionally cause students to struggle in math class?
While there are probably many reasons for inserting productive struggle (aka productive failure) into our math instruction, I will briefly share two reasons that have been on my mind lately. One is a social reason and the other is based on research.
First, the social reason for productive struggle.
Imagine a high school science class, perhaps physics or chemistry. It is an entirely common experience for the teacher to provide the students with some sort of lab experience, in which students create a hypothesis, conduct an experiment, make observations, and then write a reflection.
What is the purpose of students doing the lab experiment? Is it because we are genuinely curious about the results?
No! The teacher knows darn well what the results of the lab experiment should be. The purpose of the lab is to allow students a few moments to experience what it is like to be a scientist. To make a hypothesis. Carry out an experiment. Make observations. Reflect. The science teacher intentionally causes students to productively struggle because, for those moments, the teacher is allowing students to have an authentic experience of what it is like to be a scientist. Perhaps some students might enjoy THIS type of science enough to consider going into science as an adult.
Why then don’t we do the same thing in mathematics? Why is it entirely UNcommon for students to experience a few moments of productive struggle where they grapple with a math problem, explore solution methods (including some dead ends), make observations and reflect on their experience? In moments like this in which students are productively struggling, the REAL purpose is for students to experience – if only for a moment – what it is like to be a mathematician. Students play the role of someone who is genuinely seeking a solution or searching for a pattern rather than just replicating a rote skill the teacher has taught them.
Perhaps in these moments students might enjoy THIS type of mathematics enough to consider becoming a mathematician.
A challenge, however, is that very few teachers (me included!) know exactly what a day in the life of a mathematician looks like. Only about 50% of all 712 math teachers in California have a degree in mathematics. Even fewer have any experience being a mathematician in the real word: cryptologist, applied mathematics, actuarial sciences, etc. At best we know what it is like to be a math student, but if we want students to spend even the briefest of moments BEING A MATHEMATICIAN, we will need to learn what that means for ourselves.
Now the second reason for productive struggle…the research.
It turns out that when students are given an opportunity to wrestle with math problems to find their own solutions PRIOR to receiving instruction, their comprehension nearly doubled, compared to students learning via direct instruction only. This learning boost exists even when students are ultimately unsuccessful in finding their own solution method prior to receiving the direct instruction.
The productive struggle does more than just expose students to what it is like to be a mathematician…it also increases their ability to learn the rote skills we want them to learn!
Productive struggle is not a waste of time…in fact it ACCELERATES learning. Here is a growing collection of information about productive struggle (sometimes called “invent to learn”) that I have curated.
This presents a challenge for us teachers who were most likely trained to remove struggle, not create it. What does a lesson look like that incorporates productive struggle?
Here are some discrete steps for you to try that will incorporate both productive struggle AND direct instruction into every lesson:
 Introduce the story problem to students
 Students work independently on the problem at their desks.

 Students may use manipulatives, pictures, and/or number sentences to represent the problem.
 While students are working, teacher “teaches between the seats” to support students as needed
 (REMEMBER: it is not important for every student to be successful with this problem.)
 Select two or three students to share their thinking.

 Strategically sequence the order in which the two or three students will share.
 Arrange the student solution methods from left to right on the whiteboard.
 Allow the class some time to compare and contrast between the various solution methods
 The teacher now teaches a minilesson to show students what they will try to do in the NEXT problem.

 Connect the studentgenerated solution methods to the lesson objective
 Be explicit about what you want students to try on the next problem.
 Introduce Problem #2 for students to work on.

 This cycle continues for Problem #2 and possibly Problem #3 (if needed):
Introduce problem → Students work independently → Students share → Teach minilesson
 This cycle continues for Problem #2 and possibly Problem #3 (if needed):
Here is a visual I use as a mental model…
Here is the TL;DR on productive struggle in math class
Math teachers must learn how to create lessons that strategically incorporate productive struggle for two reasons:
 It provides an opportunity for students to experience what actually happens in the day of the life of a mathematician…much like science class lab experiments expose students to what it is like to be a scientist.
 The growing body of research clearly shows that productive struggle is an essential prerequisite for improving the effectiveness of the teacher’s subsequent direct instruction.
Not sure how to do any of this? Have further questions? Please contact me at Duane Habecker dhabecker@mcoe.org
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This is a good explanation of the desire of many of the math faculty I have been associated for their students and children. Your words better arm those who are receptive to this reform, but most administrators, parents, children, and politicians are not receptive to reform. Many decry the state of our students knowledge in mathematics but as long as their learning is so tightly adhered to their standardized grades and as long as those grades are so tightly adhered to money, I feel that this reform is futile in our public schools.
I am a Socratic instructor, which means that as a tutor, I have no students and as a professor, my students think I am too hard for no reason. Getting the right final answer to get the grade and “move past” the material is what is important. Their experience in their High School education reinforces this idea again and again. Myopic as it is, the tricks, the algorithms, and the race to circled answer are efficient and optimal… But it doesn’t optimize what you and I think are important.
We are in agreement! For students to understand mathematics and for them to experience the true joy that is inherent in learning mathematics, students need ample opportunities to act like mathematicians; to experience the the thrill of exploring the edge of their current understanding on their way towards making a breakthrough in learning a new concept.