How Scaffolding Instruction Leads to More Student Learning in Math

At the end of my first year of teaching middle school math, the most powerful thing I’d learned was that trying to teach something quickly was worse than not teaching it at all. Every lesson I tried to hustle through or cut questions from to make up for lost time was a disaster. Students ended up confused, I had to reteach rushed lessons, and what I hoped would be haste was just waste. We ended up doing the problems and activities I’d trimmed from the original lessons and then some.

My impulse to hurry is probably familiar to most teachers. Grade levels have a full year’s worth of learning packed into their standards, but there are inevitable obstacles to addressing all that content. You’re certain to have students with incomplete learning to address. You know you’re going to lose some class time due to mistakes—like a mistaught or badly paced lesson—or random bad luck (e.g., I had two weeks of jury duty in the fall of my first year of teaching.). Given these issues, it makes sense to try to be as efficient as possible. But if my bare-bones, hurried classes made things worse, what would make things better?

Scaffolding!

How Are Scaffolds Used in Math Class?

Scaffolding is the inclusion of temporary supports to help students access and complete grade-level work. Examples of scaffolding include asking questions that guide students’ thinking, giving students simpler versions of problems before introducing more complex versions, giving students a worked example, preteaching vocabulary, and breaking learning content into smaller pieces. Which type of scaffolding is best depends on the teacher, students, subject, and specific material being taught. Because of the wide variety of scaffolding strategies and the bespoke nature of their application, scaffolding can look very different from one class to another. What is always true, however, is that it involves more questions—not fewer.

On its face, it sounds like adding questions to scaffold a lesson should make it go slower, not faster. More problems, more time, right? Well, that doesn’t seem to be true. In The Great Courses^® lecture series, The Philosopher’s Toolkit: How to Be the Most Rational Person in Any Room, Patrick Grim, Ph.D., a professor at State University of New York, Stony Brook, brings up two tantalizing scenarios in which scaffolding made problem solving faster overall. First is the Tower of Hanoi: a classic math game (or math problem, depending on your perspective) invented by Édouard Lucas in 1883. In it, there are discs of different sizes stacked on three rods. The problem starts with all the discs on the left-most rod with the goal of getting all the discs on the right-most rod. There are two rules: 1) You can only move one disc at a time. 2) You can’t put a larger disc on top of a smaller disc. If you haven’t solved the Tower of Hanoi before, take a few minutes to try it.

Dr. Grim recommends that rather than trying to calculate the minimum number of moves it takes to solve the puzzle with five discs, it would be better to first solve the puzzle with two discs, then three discs, and so on. Solving simpler versions of the problem reveals the pattern needed to efficiently solve more complex versions of the problem. The application to teaching seems clear: Using scaffolded problems allows students to learn underlying concepts that they can apply to more difficult problems, which makes solving them and learning more efficient.

Dr. Grim’s next example appears to speak to this directly. He references a study by Dr. Johnathan Sweller, in which students were either given a complex problem to solve or two less complex versions of the problem leading up to the same ultimate task.

Complex Problem Only:

1. Transform the number 8 into 15 in exactly six steps. In each step, either multiply the value by 2 or subtract 7.

Scaffolded Version:

1. Transform the number 8 into 9 in two moves. In each step, either multiply the value by 2 or subtract 7.
2. Transform the number 8 into 11 in four moves. In each step, either multiply the value by 2 or
   subtract 7.
3. Transform the number 8 into 15 in exactly six steps. In each step, either multiply the value by 2 or subtract 7.

On average, the participants who got only the complex problem took more than five minutes to solve it. Those who received the complex problem after the two simpler problems needed only 90 seconds to solve it on average, with an average of three minutes to solve all three problems.

Both pieces of evidence, and my personal experience, point to scaffolding as a method of teaching that makes learning faster. As I’ve learned, however, in reading source material for this article, this is not uncontroversial, and the debate over the best way to promote learning is far from over. Because the plural of anecdote is not data, I was happy to find a recent meta-analysis (i.e., a study that examines the results of a large group of studies to try to find trends or general results) on this topic. In Meta-Analysis of Inquiry-Based Learning: Effects of Guidance, Dr. Ard W. Lazonder and Dr. Ruth Harmsen studied the impact of guidance—types of actions or supports teachers would recognize as scaffolding—on learning outcomes. They found that guidance did improve learning outcomes, supporting the other studies I read and my experience in the classroom. Because cognitive science is complicated (and education research doubly so), I can’t draw the neat “scaffolding is always faster, no matter what” conclusion that I’d like to from what I’ve read. Science just doesn’t work like that. But as a layperson, albeit a reasonably scientifically literate one, the evidence I’ve read points to scaffolding as a consistently positive, effective strategy.

So what does this mean for classroom teachers? What I’ve read suggests to me that George Pólya’s opening statement from his 1945 book, How to Solve It, still holds true:

“One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles.

The student should acquire as much experience of independent work as possible, but if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student.

The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.”

In other words, keep doing what you already knew was good for your students’ learning: using the prompts in the Teacher Resource Book, applying the On-the-Spot Teaching Tips, and supporting students without doing the thinking for them.

The Great Courses^® is a registered trademark of The Teaching Company, LLC.

Ready Mathematics helps students access grade-level work through strategic scaffolds and teachers access a rich classroom environment in which students at all levels become active, real-world problem solvers.

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About Kate Gasaway

Kate Gasaway taught middle school math at the Neighborhood House Charter School in Dorchester, MA for six years before joining Curriculum Associates. Kate’s professional experience includes writing assessments, analyses, and blog posts for Match Fishtank, an education company that creates and disseminates open-license, standards-aligned curriculum. She holds a bachelor’s degree in psychology from the Georgia Institute of Technology and a master’s degree in effective teaching from the Sposato Graduate School of Education. Kate is passionate about researching how students learn and tries to use her math powers for good.