I passed the fraction test back to my pre-algebra class and an awkward silence filled the room. The 20 students in my class received a wide range of grades, causing a mixture of satisfaction and dismay. A quarter of them had mastered the skills and were ready to apply what they had learned about fractions to solving linear equations. About half showed competency with the skills, but were still making numerous errors. The rest were lost and stuffed the tests into their binders immediately after glancing at their grade.

I would integrate repeated practice for the fraction skills into the rest of our content and I would set up a time to meet with students who had failed, but I knew it wouldn’t be enough.

The pressure of keeping up with our district’s curriculum map, getting through all the standards and preparing these struggling math students for Algebra I, led me to utter the worst possible sentence in that moment: “Time to start our next unit: Percents, Proportions and Ratios.” My heart sank.

High school math curriculum is bullet-pointed into units, chapters, subheadings and indicators. Given the massive amount of content and the tight calendar, the pace is fast—too fast for some. That’s always frustrated me, but my aggravation with the pace and compartmentalization of math curriculum was magnified by my renewed focus on the strategy of spaced practice, through my work with Dr. Yana Weinstein-Jones, a learning science researcher.

Dr. Weinstein-Jones and I began collaborating last fall as part of a project initiated by The Learning Agency, an organization that promotes improved learning through research-based methods. For the project, classroom teachers are paired up with a learning science researcher and together, each duo identifies a practice to experiment with and shares out about implementation. We decided to focus on spaced practice.

Spacing is one of the simplest learning strategies. The idea is that it’s more effective to learn something new for say, four 15-minute sessions over the course of a week, than it is to cram it into a one-hour session. Common sense, right? The research behind it is solid. Every college student has been advised not to spend an all-nighter cramming for the big exam (though most have ignored that tip at least once).

I’ve understood the principal of spacing for years and informally apply it in my day-to-day teaching, however, my work with Dr. Weinstein-Jones led me to reevaluate my implementation and wonder how I could go about measuring it.

## Optimizing a Unit for Spaced Practice

In my small, rural public school in Maine, many of our students are facing poverty and few have college-educated parents. There aren’t many local jobs for college graduates, which is one of the reasons why the majority of our students are not college bound.

I teach five math sections at the honors or AP level, and two pre-algebra sections, in which nearly all of my students are struggling to meet proficiency. I needed to find the right class to start implementation; my AP calculus class was the strongest fit. Students in this class are masters of our existing education system, which blocks content into discrete, unconnected units. They can learn a complicated topic for a short period of time for the unit test, but they are less masterful at retaining that understanding and transferring it to other situations.

I hoped that an intentional spacing approach would lead to a more durable comprehension. I’ve been teaching most of these students for several years and they were happy to experiment with me, particularly a few of them who are heading to college with aspirations to become teachers themselves.

We were about to begin our next unit, optimization word problems, which was very well-suited for this investigation, since unlike many topics in high school math, it was completely new to these students. Optimization is an application of the tools of calculus to find the best solution with a given set of constraints. Here’s a classic example of an optimization word problem:

Applying what I had learned from Dr. Weinstein-Jones, I set out to gather evidence of how I was spacing practice for this unit. Ultimately, what I found led me to redesign aspects of the unit plan to optimize it for spaced practice.

To start, I enlisted a subset of my students to help me collect some informal data. I selected four responsible students from the class and asked them to record every time they did anything related to optimization on a spreadsheet, so we could develop a visual display of spacing over time. The majority of these occurrences were in class, but I was most interested in building a record of the time they worked on or thought about this type of problem outside of class. How good of a job did I do at spreading their learning across several weeks and how much did they naturally space their own studying?

Once I recruited the students to self-report their work and the data started coming in, I began considering what changes I could make to my spacing and how that would alter the unit plan. I didn’t want this investigation to result in the same practices—lecture, homework, independent practice and quiz—just over a longer period of time. Based on my work with Dr. Weinstein-Jones, I knew the effectiveness of spaced practice relies not only on intentional spacing of exposure to a topic over time, but also on students engaging with a topic in a variety of learning modalities, so I made two significant changes.

### Implementing a Hands-On Project

First, I implemented a hands-on project early in the unit. In teams, students were given an 8 inch by 10 inch sheet of rigid plastic and tasked with building a container to hold a maximal amount of sand. Each team’s success relied on a combination of calculus skills and tactile trial and error. Each of the five teams approached the problem with a unique solution and there were many iterations and mock-ups before final containers were constructed and filled with sand.

This group challenge caused students to think about optimization outside of typical math time. Several students stopped by my classroom throughout the day to ask questions about the project or get my feedback on an idea. Dr. Weinstein-Jones hypothesizes that a significant aspect of spaced practice is providing learners with opportunities to consider an idea informally, and that when teachers carve out structured times for learners to think about a new idea in class and encourage them to think about it on their own, the understanding is notably strengthened.

The project culminated with each team filling their containers with sand, measuring the volume and creating a poster to display their solution. The sand-filled containers with the accompanying solutions are prominently displayed in the main hallway of our school and as students walk past, they’re reminded of optimization one more time.

### Replacing Traditional Quiz With a Whole Class Challenge

Typically, when teaching this unit, I assess learning through a traditional mid-unit quiz, with students working individually for 20 minutes to solve one or two optimization problems.

To shake things up a bit, I took a more conversational and collaborative approach to assessment, trying out a whole class challenge instead. The class was given 12 problems and had to split into small teams to solve and check their solutions. At the end of the period, every square inch of white board was filled with student work.

By changing the format of the quiz, I created a more durable memory for students. They’re much more likely to remember a group challenge than a run-of-the-mill independent quiz.

Throughout the unit, I built a column chart to graphically display my approach to spacing for the unit. The chart shows the times we worked on optimization during class as well as moments when students worked on their own either during homework or independent studying.

This investigation led to a reinvigorated unit plan, but more importantly, it now informs my curricular planning and it has raised some big questions, which I now consider every time I start a new unit:

- Which skills from previous units can I integrate to improve retention?
- How can I space out the learning in this unit to combat the compartmentalization inherently promoted by curriculum designers?
- How can I enrich the learning experience so that students are naturally thinking about the topic outside of classroom and during homework time?

While this work helped me understand more about my practice, and improved my student's engagement and ability to talk about optimization outside of math class, it left me with a lingering question: How can I integrate optimization into my subsequent units so that it’s not a “one and done” situation, but rather is connected to our entire study of calculus? And, of course, this question is broader than optimization and calculus—it applies to everything I teach.

My hope is that this experience will guide me in moments when I find myself saying that dreaded line, "time to start our next unit,” by helping me find ways to integrate and effectively space skills from one unit into the next to provide more opportunities for kids to practice what they’ve learned.