# Media and Education in the Digital Age

## Concepts, Assessments, Subversions

## Edited By Matteo Stocchetti

# The Future of Mathematics Textbooks: Ramifications of Technological Change

# The Future of Mathematics Textbooks: Ramifications of Technological Change

Abstract

As mathematics educators, the object of our research is a societal endeavor whose policies and practices are shaped by societal forces, including technological developments. Textbooks have historically played key roles in determining the mathematics curriculum by specifying the content to be taught and by providing guidelines about how this content might be taught. In this paper, we argue that technological changes pose challenges to the roles played by the textbooks and curriculum materials written by textbook authors and curriculum developers. The role of specifying what is to be taught is under challenge from centralizing forces supported by technological capacities for large-scale data mining. And, the role of providing guidance on instruction is under challenge from changes to processes for authoring and publishing books; these changes have the potential to shift the role of teachers in the curriculum development process. While we do not see these challenges as representing the death knell for textbooks, we argue that with these technological changes, textbooks may no longer play as large a role as a driver of educational change. To support our argument, we explore the historical roles of mathematics textbooks in educational systems and specify two challenges, supported by recent technological advances, to these roles that we have outlined above.

“I predict that the word “textbook” will soon carry the same connotation as the word “scroll” does today. The word “scroll” harkens to a time when scholarly materials were produced on papyrus and stored in cool vaults, to be accessed by the few elite lucky enough to have access. Likewise, with the proliferation of scholarly works on the Internet, students and teachers are now able to create their own repositories of knowledge based upon reliable sources. Some classes are eliminating textbooks all together, opting for sites like Curriki or creating projects where students compile, authenticate, and create information based upon their research.” (Davis, V. downloaded from http://coolcatteacher.blogspot.com/2009/06/are-traditional-textbooks-dead.html)

While many mathematics educators throughout the world^{1} would agree that in the present in K-12 education “Textbooks represent a substantial financial investment ← 63 | 64 → and strongly influence what students learn” (Reyes, Reyes, & Chavez, 2004, p. 61) and, thus, are legitimately an important focus for international comparisons of students’ opportunities to learn mathematics (e.g., Haggarty & Peppin, 2002), the blog-post quoted above raises many questions: As a part of larger effects of technology on education (Christiansen, Johnson, & Horn, 2008; Collins & Halverson, 2009; Cuban, 2001; Ferneding, 2003; Selwyn, 2011), in the developed world,^{2} will mathematics textbooks endure? Might other educational products come to play a more central role in communicating the mathematics curriculum than textbooks? Will mathematics teachers, in addition to textbook authors and curriculum developers, play a more central role in creating curricular materials for students? In brief, our responses to these questions are that mathematics textbooks will endure, though they may evolve and their role in the education system may become less central because common standards and assessments will play some of the roles that textbooks used to play. And, we expect that though teachers may be able more easily to edit and author documents, they will probably not take the lead in writing textbooks. The purpose of this chapter is to support those speculative answers by exploring the historical role of mathematics textbooks in educational systems and identifying challenges, supported by recent technological advances, to these roles.

# Mathematics textbooks in state^{3}-provided education: A historical perspective

As argued by Kidwell, Ackerberg-Hastings, and Roberts (2008), the mathematics textbooks that are so ubiquitous in schooling now were once, like public schooling itself, non-existent. Once people learned advanced mathematics by reading original mathematical texts, like Euclid (Herbst, 2002), with a tutor. When schooling became institutionalized, supported by the technological changes to printing, students began to study mathematics with teachers and grade-level textbooks provided at no cost by the school^{4}. ← 64 | 65 →

Historically, textbooks developed in compulsory, state-provided education as a mechanism both for outlining and specifying what it is that students should learn and in what order (Westbury, 1990). In addition, textbooks provide teachers with guidance about how students should learn that content by providing exercises and exposition, though teachers are meant to use their professional judgment in determining how to use and supplement a text, both with activities and with other resources.

Mathematics educator Gert Schubring (1987) describes, as a turning point in the development of books created as textbooks, the time not long after the French Revolution. He suggests that at this time there was an attempt to democratize access to mathematical knowledge; that the early nineteenth century was:

… an epoch when for the first time a general and public system of education became established … This new educational system gave mathematics a considerable position… The structure of the presentation of mathematical knowledge was not prepared and adapted for the demands of this sort of teaching. A restructuration and redefinition of mathematical knowledge became necessary. (pp. 42–43).

No longer could mathematics be diffused solely through the correspondence of individual scholars (see Pearl, 2010, for example, on the role of Mersenne’s correspondence in the mathematical advances of his time), the work of mathematicians, such as the Frenchman, Lacroix, in writing textbooks helped restructure mathematics in ways that made it suitable for this evolving educational system.

The development of these texts for general education in France in the early 19^{th} century, like the later work of Bourbaki to systematize modern mathematics (Borel, 1998; Guedj, 1985), were influential throughout the world. For example, Charles Davies at the United States Military Academy at West Point helped found the nascent US textbook industry by translating French mathematics textbooks and eventually providing books to schools, as well as colleges (see Kidwell et al., 2008: 10–20). These textbooks supported the development of the mathematics research community in the United States (Parshall & Rowe, 1997).

Skipping more than a century ahead, in the mid-20^{th} century, in the US, the mathematics curriculum is influenced by the challenge of integrated modern mathematical ideas into the pre-collegiate curriculum, concerns about pedagogy in schools, as well as continued competition between countries (Conference Board of the Mathematical Sciences, 1975). In some countries, curriculum development projects whose work is funded by the state, rather than individual mathematicians or mathematics teachers (like Lacroix or Davies), begin to produce some of the mathematics textbooks used by schools. For example, in the United States, the government, in the guise of the National Science Foundation, reluctantly, out of a fear of the political nature of curricular work (Dow, 1991), funds university-based mathematicians and ← 65 | 66 → scientists to play a crucial role in the updating of mathematics and science curriculum for schools (Rudolph, 2002, see Ch. 2). Thus, in the forward (1963) to *Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics,* Frances Keppel, then US Commissioner of Education, points out the ways in which this activity seemed noteworthy at the time:

“If one were to look for the most significant development in education over the past decade, it would be reasonable to single out the wave of curriculum reform … They [these recent curricular reforms] have been for the most part national, or at least regional, efforts. They have drawn on university scholarship and skilled teachers not only for leadership but for the immediate demands of day-to-day operation … Almost without exception they have passed from the determination of policy and program directly into the preparation of materials for use in the schools” (p. vi).

The focus on preparing materials for use in schools was premised on the belief that simply articulating policy and program would not be sufficient. In order to have new content and ideas about teaching incorporated into classroom activity, teachers would need the guidance of new curricular materials.

The US National Science Foundation then, based on confidence in the power of curricular texts, repeats this early foray into curriculum development at the end of the 20^{th} century, in concert with the NCTM Standards movement (Lappan, 2003). A division of labor evolves: Publishers and the marketplace provide status quo textbooks, while curriculum developers based at academic institutions, now working in concert with some publishers, create reform textbooks that are meant to move schools in the direction of the incorporation of both new content and new methods of teaching (in the sense in which NCTM advocates both principles of instruction and content standards in the 2000 *Principles and Standards for School Mathematics*). In this division of labor, teachers play a role in advocating for reform-based materials and implementing the vision of instruction that is concretized in curricular materials. As articulated by Glenda Lappan (1998), in her role as president of the North American professional organization of teachers, the National Council of Teachers of Mathematics, the role of the teacher is to work with colleagues to adopt these materials and then: “to increase the effectiveness of the materials by spending considerable time planning the lessons, listening carefully to what students are saying in the classroom, analyzing what students are learning, and consequently adjusting the mathematical tasks and the questions asked.” Given the realities of teachers’ work-lives, their task is not to have to search for, or create, rich mathematical tasks for their students. And, given the ways in which mathematics as a discipline grows and changes, it is not the individual teacher, or the citizenry of the country, who should make judgments about what students should learn, but people more deeply acquainted with the larger picture ← 66 | 67 → of what mathematics there is to know and how knowledge of that mathematics will prepare students for future education and careers.

# Articulating two inter-connected roles of textbooks: What and how

The purpose of this quick recounting of key points in the history of mathematics textbooks was to illustrate the roles that textbooks play in state-sponsored education. We suggest that textbooks give teachers guidance on both *what* and *how* students should learn. On the one hand, especially initially, textbooks organized the content of what students were to learn and indicated what students needed to know at what age, grade level, or institutional track within schooling. On the other hand, by presenting instructional tasks, textbooks attempt to organize the knowledge that they present in ways that will help make this content learnable (Cohen, 2011). In playing both of these roles, textbooks support teachers by allowing them to focus their attention on modifying and customizing tasks to help their students meet the articulated goals; textbooks are a vehicle for people who are not teachers to give teachers guidance both on what, and how, students should learn.

Textbooks play these roles by virtue of what they include and what they do not, as well as how they organize content into chapters and how they distribute content over grade levels. They play these roles by dividing content into sections and providing exercises for students and teachers to assess whether or not students have acquired the knowledge that is intended. Increasingly, they also provide formative assessments and other ancillary materials that will help with remediation (For other features and descriptions of textbooks in mathematics education, see Love & Pimm, 1996).

Against this backdrop of two roles that textbooks have occupied in schooling, we see two challenges to the historical role of the textbook, each of which is gaining impetus and support from changes to today’s technological environment. Our argument in the next two sections is that new technological developments potentially contribute to altering the existing balance of power between textbooks, syllabi and standards, and assessments, a balance of power that much current research assumes.

# Competition to textbooks for outlining what students should learn: Technology-supported accountability and assessment

If curriculum is “what students have an opportunity to learn in school, through both the hidden and overt curriculum, and what they do not have an opportunity to ← 67 | 68 → learn because certain matters were not included” (McCutchean, 1982, p. 19), then textbooks, as a component of education that is provided by the state, are enmeshed in the political processes^{5} that determine the intended curriculum (Seeley, 2003). Though perhaps in the early 19^{th} century textbooks were the primary means for outlining what students should learn, these days, other documents compete with textbooks for the role of articulating the intended curriculum in the educational systems of many countries (See International Review of Curriculum and Assessment Frameworks Internet Archive, 2009).

Additionally, assessments at key educational junctures (e.g., exams at the end of junior secondary school in China; or, exams in the US that are required for high school graduation, see Center on Education Policy, 2009) play an important role in communicating expectations about what students will learn. So, textbooks do not determine curriculum on their own, rather they are a part of a system that produces the experiences students have in school. Teachers play an important role in shaping the curriculum students experience, and larger political processes shape the intended curriculum.

In our view, the rising influence of standards and assessment has a technological component. The rate of the accumulation of digital data has skyrocketed and there are robust developments in technologies to “mine” such data (as the very coining of this phrase suggests). For example, as documented in the book, “Total recall: How the E-memory revolution will change everything” (Bell & Gemmel, 2009), the MyLifeBits project at Microsoft Research (http://research.microsoft.com/en-us/projects/mylifebits/) has explored the technological changes that support collection, storage, and searching of data:

“We are capturing so much of our lives now, be it on the date- and location-stamped photos we take with our smart phones or in the continuous records we have of our e-mails, instant messages, and tweets – not to mention the GPS tracking of our movements many cars and smart phones already do automatically… the critical technology and perhaps the least understood, is our magical new ability to find the information we want in the mountain of data that is our past. And not just Google it, but data mine it …” (Front cover).

Increasing capacities to collect, store, and search data in the developed world have influenced aspects of social policy such as community policing, and also are relevant to the data accumulated by education systems (Mosteller & Boruch, 2002). New data systems may make possible innovations in the assessment of teachers, ← 68 | 69 → schools, and districts. For example, in the last decades scholars have begun to explore value added techniques for determining what part of the variance in student achievement may be attributed to schools or teachers, rather than to family or socio-economic status (McCaffrey et al., 2003; 2004).

Without delving into the controversies about these statistical methods (Baker, Barton, Darling-Hammond, Haertel, Ladd, Linn, Ravitch, Rothstein, Shavelson, & Shepard, 2010), these techniques require common curricula for students, common assessments given to students each year, as well as data systems that provide each teacher and student with a unique identification number that moves with them and that store student results and link them to teacher identifiers.^{6} The appeal of such data systems is in enabling governments to hold educators accountable for results; proponents of the development of a greater national commonality in mathematics curriculum, see a range of benefits, including greater focus in textbooks (in the US context, see American Educator, 2010). Critics of these notions of accountability suggest that such developments are not to be applauded, that the requirements of data systems will lead to homogeneity in curriculum (e.g., Ravitch, 2010). The implication for textbooks is that their role in outlining *what* students learn might diminish; textbooks would need to reflect what other artifacts of institutional schooling outline must be taught and learned. Increasingly, textbooks or curricular materials for the same grade level or course would all cover the same content.

# Technologically-supported changes to the role of teacher in preparing materials for instruction

In addition to an explosion in the accumulation of digital data, advancements in technology have also changed the nature of texts, as well as what it takes to produce and disseminate information (See Young, 2007, for a critical analysis). Who authors text materials, what it means to publish a text, and the speed at which ← 69 | 70 → texts are updated are all undergoing shifts, all of which have implications for textbooks. With textbooks, these developments suggest changes to the traditional relationships between textbook author or curriculum developer and teacher.

The role of the teacher historically has been to shape the textbook for use with particular groups of students in particular places on the basis of the teacher’s own professional judgment. Technological developments change what it means to supplement a textbook. In the past, supplements were done with dittos and Xerox copies; now, such curriculum supplementation can be done with digital texts written (and easily edited) by the teacher, or fetched from the web [in what some might call a web 1.0 style].^{7} And, as digital composition, publication, and production become more common, beyond activities and software, these supplements can include teachers using Smartboards to add records of the work done in their classes, snapshots of the board, audiotape of classroom discussion, or perhaps even edited digital video recordings of class.

With these kinds of technologies, supplementation begins to get closer to an “open culture” concept of co-authoring a new version, rather than supplementing an existing version that remains unchanged. According to an “Open Culture” perspective, particular knowledge products or texts are not fixed; knowledge should spread freely and its growth can come from developing, altering, or enriching already existing knowledge products on a collaborative basis, without being restricted by rules linked to the legal protections of intellectual property. Licenses, like the Creative Commons License, stipulate that any product that uses a Creative Commons License resource as an input must be returned to the commons and cannot be subject to copyright. In line with this ideology, “Open Educational Resources” – web resources that can be used according to flexible copyright licenses – are created by teachers working iteratively to refine and edit the work of other teachers and posting the products of their efforts on the web for use (Morris & Hiebert, 2011, offers a related vision of teachers creating shared instructional products, but does not address issues of license).

For example, Neeru Khosla, the founder of CK12.org, a digital textbook publishing environment for schools, suggests that teachers might use the CK12 platform to create personalized parts of a book to support the learning of students with disabilities; she then uses the term “Flexing” to describe the process of adapting an existing digital book to turn what was initially created as a personalized resource ← 70 | 71 → for learning into a text that others can use (See http://www.educationnews.org/michael-f-shaughnessy/47145.html). Similarly, an Israeli site (http://www.school.kotar.co.il/Default.aspx) envisions a setting where students get a paper textbook and access to the digital format of the same book including links to interactive resources provided by the publisher. The site then encourages teachers to add their own web resources and “post” them to replace specific activities. Connexions.org is a similar site that provides users with a place to view and share educational material. With this site, authors can build up modules out of small knowledge chunks that can be organized as courses, books, or reports.

With such sites, groups of teachers coalesce initially around materials developed by others or an authored textbook. But, perhaps, over time, influenced by open culture ideas and sites like Connexions.org, as groups of teachers gain experience amending or sometimes personalizing digital textbooks, they might decide to develop textbooks as a communal endeavor. Such communal creation of products is a part of what people now associate with Web2.0 (For one set of defining principles, see http://oreilly.com/web2/archive/what-is-web-20.html), Wikipedia being a prominent example of the kind of new process of authoring that can be done with the networked technological tools now at our disposal. Wikipedia provides users with tools to write, edit and change entries; at the same time, previous versions and changes remain visible to the public. With Wikipedia, there is a commitment to a shared style of writing that makes the result a well-integrated material. Contributors strive to take a neutral view and to include sympathetically a variety of views, rather than to present a single commonly accepted as the objective truth. There are authorities that may exclude inappropriate material, but those authorities rarely use their power. The question for open textbooks initiatives (e.g., www.wikibooks.org) is whether Web 2.0 principles can be successfully integrated into a system that will have large numbers of teachers and other stakeholders collaboratively creating textbooks that are widely used.

How do the implications of the technological support for this kind of creation of public personalized versions of knowledge products sit alongside the trends to curriculum centralization outlined earlier? Perhaps, if the push toward centralization and common standards continues to gather momentum, teachers will choose to coalesce as groups around standards or curriculum documents, rather than around textbooks. This possibility is plausible especially as curriculum and standards documents flesh out their imperatives with examples meant to illustrate their intent. Thus, the NCTM provides e-examples along with its Principles and Standards (NCTM, 2000) document; the Common Core State Standards Initiative is contemplating a product that provides examples of tasks for classroom use; and in centralized systems curriculum documents are moving beyond syllabi to ← 71 | 72 → include examples that teachers download and use in their classrooms. For example, in Israel, the ministry’s mathematics supervisor maintains a site with sample lessons for a wide variety of tasks by grade level. Thus, in addition to a relatively short document outlining “what to teach,” the ministry provides digital resources that include “how to teach.”

# Justifying our initial speculations

Stepping back from our descriptions of the ways in which technologies support both teacher involvement in the processes of creating and supplementing textbooks and the roles of technologies in the current push toward the creation of common curricula and assessments, we now return to justify our speculative assertions that:

- Textbooks will endure, though they may evolve;
- Teachers will probably not take the lead in writing textbooks; and
- The role of textbooks in the education system may become still less central because common standards and assessments will play some of the roles that textbooks used to play.

In suggesting that textbooks will endure in the face of recent advances in technology, we find compelling Yochai Benkler’s (2005) analysis of the differences between textbooks and encyclopedias. At the heart of his argument is the notion of modularity. He argues that:

“Real textbooks appear to reside somewhere between a novel and an encyclopedia in the degree to which they can be modularized, or at least in the degree of effort required to integrate the modules into a coherent whole recognizable as a textbook. Moreover, the chunks or modules seem to be bigger. It is very hard to add a single sentence, although it may be possible to add a single example or a better-rendered equation or chart” (p. 20).

He uses these differences to argue that, even though Wikipedia has been successful, we will not see web 2.0 textbooks as dominant force in the future.^{8} His key point ← 72 | 73 → is that the lack of modularity in textbooks and their “need to be systematic, coherent, and compliant with well-defined external constraints” (Benkler, 2005: 22) intrinsically makes them a kind of document that is not as conducive to peer-creation (For an examination of this question, see Yerushalmy, 2011).

Of course, there are many who disagree with Benkler’s conclusion, for example, Vicki A. Davis the teacher at Westwood Schools in Camilla, Georgia whose post opens our paper. Nonetheless, in her post, Davis conceptualizes a continued role for textbook companies in the future as “content conduits.” She imagines that textbook companies will continue to have a role in making sure that experts can communicate with teachers, even if the nature of their textbook products will evolve. In her view of this future, the role of the teacher will not be to create the core text, but to augment it and connect it to lesson plans, student work, and more. Perhaps imagining a more central role for teachers seems impossible without some fundamental changes in the work-life of the teacher; the daily pressures of time and face-to-face interaction with students may limit visions, even of teacher bloggers, for the participation of teachers in the creation of curricular materials.^{9}

If these observations are correct and textbooks and curriculum developers do not have to fear replacement by wikibooks written by teachers, then, perhaps textbooks and curriculum developers may face a graver threat from examples and illustrations coalescing around mandated standards. Interestingly, in the context of debates about how geometry should be taught in the Israeli middle grades, it was a group of Israeli mathematicians who were concerned about this possibility (See http://www.haaretz.com/news/national/professors-call-israel-s-junior-high-math-program-scandalous-1.330105). They argued that the curriculum documents on the ministry’s website reflected a specific opinion regarding the order the unpacking of the geometrical axioms and how to teach geometry; their criticism was grounded in the view that the order of the development of ideas, as well as the examples and sample lessons, should remain the province of textbook authors and should not be provided by the Ministry of Education’s committee that articulates the content of the curriculum. While their argument, and its assumption that it is possible to disconnect *what* is to be taught from *how* it is to be taught, supports ← 73 | 74 → an ongoing role for textbooks that is different from that of documents articulating standards, the very necessity of making the argument suggests that in a web-based environment, there are dynamics that will lead curriculum standards to encroach on what has in the past been the domain of textbooks.

Assuming for the moment that our predictions are well-grounded, are the changes to the roles of textbooks that we have described consequential? We close with one concern about the impact of current trends on the curriculum development process. As our earlier description of the important roles played by textbooks and curriculum developers in earlier eras of mathematics reform suggests, textbooks have played a role in the evolution of curriculum change over time. If textbooks and curriculum materials are tightly coupled to standards and have little leeway to experiment, particularly with new content, what remains unclear in this age of centralization is how processes of curricular change will happen. How will the mathematics curriculum grow and change to reflect evolutions in mathematics itself? How will competing curriculum visions be outlined in enough detail to understand the potential they have as alternatives to the curriculum as outlined in common standards?

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1. Our professional experiences have been mostly in the United States and Israel and as a result our citations and references are skewed toward experience in these countries; this paper is not an attempt to do cross-cultural work in education. While we know that there are many important differences in the nature of mathematics education in different countries, as Leung and Li (2010) observe “The increased ease of flow of information through exchange activities as well as through the internet has meant that [educational] systems are increasingly influencing each other” (p. 4). For this reason, we believe that the dynamics we point out are more widely relevant.

2. We are keenly aware that in many parts of the world state-sponsored compulsory K-12 education is still an unfinished project and that textbooks, let alone digital textbooks, are not as widely available as might be preferred, but we will not continue to emphasize this point in the text.

3. We will use “state” in the sense of nation, not in the sense of a unit of organization inside a federation as is the practice in the United States.

4. Initially, in the US, the decision to provide textbooks at no cost was controversial, see Westbury, 1990, “Massachusetts becomes the first state to require all communities to provide free textbooks for their students. This notion of free textbooks provided by the school, and the related “adoption” or “approval” of these texts became American institutions” (p. 7).

5. Apple, 1989, for example, articulates how in the United States, tensions between the North and the South after the Civil War, and the identification of public schooling and textbook publishers with the North, lead to the policy of state-wide textbook adoptions by southern states, rather than by individual districts within these states.

6. The creation of a technological infrastructure for such comparisons within states in the United States forms a substantial part of the investment of the US Department of Education’s recent “Race to the Top Fund” initiative which “provides competitive grants to encourage and reward States that are creating the conditions for education innovation and reform” (Downloaded from http://www2.ed.gov/programs/racetothetop/index.html on January 4, 2011). Two of four “core education reform areas” in which states must implement ambitious plans are “Building data systems that measure student growth and success… [and] Recruiting, developing, rewarding, and retaining effective teachers and principals …” (U. S. Department of Education, 2009, p. 2).

7. Sites that are used in such ways include: illuminations.nctm.org, geogebra.org, or mathforum.org/mathtools/. These sites, and others, show evidence of teachers investing time and effort both in the development and sharing of materials that reside on the web for use by others and in the use of materials developed by others.

8. Bolstering his argument empirically, in reviewing existing wikibooks, he notes that there are not many textbooks and that the work, in general, is not of a large community; most wikibooks seem to have a single author, sometimes with minor contributions from a small group. These trends seem to be continuing, among the textbooks listed under k-12 mathematics wikibooks [retrieved January 26th 2011], there is only one featured book which is authored by a single main author and three contributors, students who edit the mathematics and the solutions. Similarly, the texts at CK12.org, which are the only mathematics textbooks listed currently under the State of California initiative (http://www.clrn.org/FDTI/index.cfm), do not have large authoring communities.

9. We are not suggesting that teachers will not blog and write about their teaching; we expect that as blogging becomes even more ubiquitous more teachers will be drawn to such outlets. But, even though there are experiments at converting blog into texts, we do not see blogs becoming textbooks. For provocative experiments in this direction, consider: http://booktwo.org/notebook/wikipedia-historiography/ and http://booktwo.org/notebook/vanity-press-plus-the-tweetbook/.