Give Students a Moment to Be Mathematicians

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 7-12 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:

1. Introduce the story problem to students
2. 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.)

3. 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

4. The teacher now teaches a mini-lesson to show students what they will try to do in the NEXT problem.

• • Connect the student-generated solution methods to the lesson objective
  • Be explicit about what you want students to try on the next problem.

5. 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 mini-lesson

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:

1. 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.
2. 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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