Recently, the 60,000-member National Council of Teachers of Mathematics (NCTM) released a new “Research Compendium,” a handbook with 38 chapters, each summarizing the best research on an important aspect of teaching and learning mathematics. For the past three years, I was honored to work on the team that developed the chapter on “Technology for Mathematics Learning.” The aspects addressed in other chapters included summaries of how students learn specific mathematics content and practices, research on teaching, and much more. Writing the technology chapter gave my team the opportunity (and challenge) to comprehensively organize the vast research knowledge about technology in learning mathematics. So what did we discover?

First, humility! As my mentor Jim Kaput vividly wrote in a 1992 NCTM research handbook “anyone who presumes to describe the roles of technology in mathematics education faces challenges akin to describing a newly active volcano.” That volcano is still erupting; the geography of digital possibilities for math learning is constantly expanding.

With co-authors Richard Noss, Nicholas Jackiw, and Paulo Blikstein, we quickly realized that if we tried to pen a consumer guide to research on each of today’s available mathematics technologies, our findings would be obsolete before the ink dried. Indeed, the chapter was finalized before the results of my own most recent efficacy trial, on the use of ASSISTments for online mathematics homework, could be included. So this is not a listing of product-by-product research, but rather an organization of research about categories of digital mathematics tools.

Second, the value of a well-chosen framework. We chose to organize around a framework developed by Paul Drijvers at the Freudenthal Institute in the Netherlands. Drijvers identified three main ** purposes** for technology in mathematics learning:

**Doing Mathematics.**Tools for this purpose are used both in and out of school, and include tools like calculators and spreadsheets. Such tools can handle distracting details for a learner and enable a learner to focus cognitive effort on mathematical actions that are closely related to their current learning objective. For example, if you have already mastered the skills of plotting points to draw a graph, it can be beneficial to let a graphing calculator do that. Then the learner can focus an important mathematical concept, such as how the shape of the graph changes with changing function parameters.**Practicing Mathematics.**Tools for this purpose organize the sequence of a student’s practice sessions, help students when they struggle, and provide feedback. This purpose can have a positive impact on learning because working many practice problems really is an important aspect of learning mathematics. Yet in school, mindlessly filling out worksheets is too often what actually happens, and mindless mathematical grunt work is not beneficial. Consider an analogy to sports training, where science is making athletes better by shaping more effective practice sessions. As in sports, learning sciences research has uncovered much about the optimal way to organize and support mathematics practice. Today’s technologies can help to optimize the practice sequence, support students when they need help, and give well-timed and targeted feedback.**Understanding Mathematics.**Tools for this purpose help students make coherent connections among mathematical ideas that otherwise can appear to students as arcane, arbitrary and disconnected. A central learning sciences insight is that using spatial visualizations that vary in time to engage students with mathematical relationships is a powerful aid to forming appropriate mathematical connections. For example, imagine two situations in which a student is trying to understand what geometry proof means. In one situation, the student tries to infer the meaning by looking at a printed geometric figure. That’s hard. In the other, the student moves their finger and experiences how a displayed figure changes in time according to the rules of the proof. Research shows that “dynamic representations” (using rule-driven visualizations that move) can lead students to “a-ha!” moments. Thus, technology can enable new ways to represent mathematics dynamically, and this can help students make sense of math.

In preparing the chapter, we found that for each of these three purposes there is a wide selection of high-quality tools. Further, each purpose also can organize along relevant learning science principles, empirical research findings, and other insights that can help educators to use the technologies well. Importantly, we were able to find strong research support for all three purposes — it is NOT the case that one purpose is best. Thus, we’d recommend that educators start with a refined sense of their own purpose for bringing technology into their students’ mathematical learning process. Once they select a purpose, we’d recommend careful attention to the learning principles and empirical findings specific to that purpose. In our view, technology is a valuable enabling infrastructure, but it takes an educator’s concerted attention to many factors outside the technology to make a difference for their learners.

**Interest-Driven Mathematic Emerges.** In addition, the research literature pointed us to a fourth, emerging purpose — one with a smaller but growing scientific literature. Increasingly, research is studying what happens as students become engaged in technology-rich activities in robotics, fabrication labs, maker activities, electronic crafting activities, coding activities and more. In part due to the mathematics intrinsic to the technologies in these activities, there is increasing potential for Interest-Driven Mathematics. Interest-driven mathematics is different from the other purposes, because it emerges in extra-curricular settings, rather than being driven by workplace or school prerogatives. In our chapter, we recognized the need for more research to guide educators who seek to develop students’ mathematics learning as the purpose to technology-driven extracurricular interests.

**Complementary Factors: Productivity and Transformation.** The chapter enabled us to see an important pair of factors that were essential to the best research-based uses of technology — and to realize that these factors were complementary and not in opposition.

One factor is productivity. Technology in mathematics learning either scales or fails depending on whether it makes teachers’ and students’ efforts more productive. The productivity dimension can rise to the surface in a graphing tool that enables students to focus on the shape of a function’s slope instead of the tedious process of plotting points. It can also arise in a practice tool that makes homework sessions more fruitful. Alas, too often this factor can also arise negatively when teachers find that technology runs out of batteries, internet connections fail, or servers are down. When technology wastes precious classroom time, it’s less likely that use will continue. Conceptual understanding tools often offer attract teachers with their potential outcomes — for example, a shift to deeper learning objectives or a flipped classroom model. And yet an eventual “scale or fail” outcome often depends on how attentive the program is to teacher and student’s productivity as they pursue these aspirational outcomes.

The other factor is transformation. Technology in mathematics learning either attracts zeal or declining appeal not only due to productivity, but also to the degree it enables a shift to a much more desirable mathematical learning experience. The transformational factor is often close to the surface when educators apply technology for the purpose of conceptual understanding, where many initiatives have the purpose of “not only learning mathematics better, but also learning better mathematics.” In one example covered in the chapter, MiGen aims to help students participate in mathematical generalization — a fundamental aspect of professional mathematics that is vanishingly rare in conventional classroom experiences. Transformation is also inherent in the new category of interest-driven and technology-rich mathematics — it proposes the radical idea of locating math in activities that students love, rather than trying to make school mathematics “relevant” to students lives after-the-fact.

We also found in writing the chapter that the attracting zeal or fading appeal factor is important to the doing math and practicing math as well. With regard to tools for doing math, a division-of-labor view of graphing calculators and spreadsheets is shallow. It’s not JUST that a student can offload calculations to a calculator or spreadsheet and focus their effort on a mathematical strategy. The zeal comes as learners grow into a mathematical tool, and become aware of how its advanced capabilities make it an indispensable partner in their reasoning process — as the relationship of person and tool doing mathematics becomes transformed (a process of co-evolution called “instrumental genesis” in the literature). Tools that build a zealous following don’t just save effort, but also offer a trajectory of growth in mathematical capability distributed across the mind and machine. Likewise, we noted that a cognitive tutor not only manages student’s mathematical practice, but also creates programmatic opportunities to re-imagine the teacher’s role (for example, as intensively working with individual students) and to allow for more time and focus in non-technology activities, for example, collaborative learning among students. Thus, practice tools that survive for longer times in the marketplace also develop their potential for transformation.

Hence our recommendation to educators is to embark on programs of applying technology in learning mathematics with attention to both productivity and transformation, as both the fail or scale and attracting zeal or fading appeal dimension were strongly evident in all long-lasting research programs regarding technology in mathematics learning that we were able to identify.

To close, I heartily recommend the entire Compendium to all who are committed to improving mathematics teaching and learning (disclosure: my only financial compensation as an author was to get a free copy — which I definitely will be using regularly for all great information in the other chapters). Although the Compendium is a bit pricy, the one-stop-shopping it offers on such a full range of research topics in mathematics education will be invaluable. And realize that 60,000 mathematics teachers have already contributed to the production of this volume through their NCTM membership dues. It’s an opportunity to stand beside these mathematic teachers’ commitment to pulling together high quality research. Further information on the Compendium is available on the NCTM site.