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Why is Slope Hard to Learn?

With the recent release of our new middle school lessons—in which students learn about rates of change—it’s clear we’ve been thinking about slope at DreamBox for quite some time. And we’re not the only ones; Sybilla Beckmann and Andrew Izsák recently wrote an article about their current research called “Why is Slope Hard to Teach?” for the American Mathematical Society’s blog, On Teaching and Learning Mathematics.  It’s a thoughtful piece that I’d highly recommend.

Beckmann and Izsák do a great service to students and teachers by starting their post by drawing the distinction between Novices and Experts (see here for some of my own thoughts about that difference). They also frame the conversation with this important reality about learning that has been confirmed by decades of research:

“Yet experienced math teachers know that presenting mathematical ideas clearly, as important as that is, is generally not enough for students to learn the ideas well, even for dedicated and determined students.”

In that sense, the reason teaching slope is so difficult is because teachers can’t rely solely on presentation and explanation when creating lessons that will result in student learning. In their book, Best Practice, Zemelman, Daniels, and Hyde make this related point:

Presentation of an explanation, no matter how brilliantly worded, will not connect ideas unless students have had ample opportunities to wrestle with examples” (p. 170, Heinemann, © 2012).

Despite these admonitions, most educational programs and software simply digitize content presentation in ways that cannot result in the learning outcomes we want for students, such as critical thinking and strategic problem solving. Fullan and Donnelly, in their report, “Alive in the Swamp” make this point about the limitations of most existing edtech “innovations”:

Many of the innovations, particularly those that provide online content and learning materials, use basic pedagogy—most often in the form of introducing concepts by video instruction and following up with a series of progression exercises and tests” (p. 25, NESTA, © 2013).

Even worse are the print and digital education resources that try to teach math using gimmicks instead of research-based instructional strategies.  Consider this point made in a recent Hechinger Report article about teachers’ experiences vetting math resources for alignment with the Common Core State Standards:

“The Common Core calls for students to grapple with challenging math on their own, writing out the steps. So a math program that promises to teach students math by having them memorize simple rhymes? It’s probably about as legit as…diet deep fried ice cream.”

Understanding slope is critical to success in Algebra 1, and DreamBox is supporting that success for middle school students. Our teachers here at DreamBox have years of first-hand classroom experience in seeing how difficult slope is for students. And to deeply understand slope, we realize that students need much more than simply an explanation, a lecture, shortcut gimmicks, or a few worked examples with multiple-choice practice. That’s why we’ve created innovative digital experiences and an interactive coordinate plane that engages students in making sense of slope by wrestling with examples and figuring out strategies and connections on their own.

To make sure students learn slope effectively while working independently, our DreamBox slope lessons have consistent learning progressions within and across grade levels. As you can see in this overview video, DreamBox makes slope very accessible to novices while also requiring experts to find the local rate of change at a given point on a parabola. To engage both novices and experts, DreamBox teachers write these lessons to provide real-time scaffolding and feedback based on each student’s own intuitive solution strategies and answers, as you can see in this example. If you’d like to experience one of these lessons just as a student would, you can access it here.

As additional educational technologies become available, it’s important to evaluate the quality of these resources to ensure that they are creating rigorous and meaningful learning for students. At DreamBox, we’re committed to creating transformative experiences that go far beyond digitized worksheets. We’re creating lessons that can’t be replicated without digital technology, so that students can connect with important ideas in ways that aren’t possible without technology. To help evaluate and select digital curricular resources for any content area, check out my recent white paper for some ideas and insights.

Tim Hudson