To develop our teachers’ teaching mathematics in the digital age, we took careful attention to integrate technology into our professional development. The Technology Principle in the Principles and Standards of School Mathematics (NCTM, 2000) state that, “Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances student learning” (p. 24). The phrase “influences the mathematics that is taught” is what determines the ambitious teaching goals in the mathematics classroom.

The word “enhances” is what characterizes technology as a tool with high- leveraging power because technology has specific affordances that can enrich learning tasks (Suh, 2010). Niess and Walker claim that (2010) “many digital technologies have proved useful for students learning mathematics: graphing calculators, applets or virtual manipulatives, spreadsheets, computer algebra systems, and dynamic geometry tools. Each of these technologies provides visual representations that enable students to explore mathematical ideas in more dynamic ways.”

However, just by having technology in the classroom does not guarantee that technology is effectively implemented in the teaching and learning of mathematics. The complexity of teaching with technology stems from the notion that teaching in itself is a complex endeavor. Shulman (1986, 1987), coined the term Pedagogical Content Knowledge (PCK) to describe the specific knowledge needed to teach effectively that includes content knowledge, knowledge of students’ thinking, and knowledge of mathematics education and pedagogy.

In mathematics education, PCK has been expanded to include Mathematics Knowledge for Teaching (MKT), which is the “knowledge necessary to carry out the work of teaching mathematics” (Hill, Rowan, & Ball, 2005). MKT includes specific high- leverage practices such as use of mathematical explanations and representations, interpretations of student responses, and the ability to avoid math errors and imprecision.

Sequencing technology applets to reflect on the mathematics learning progressions

Figure 2.3 Sequencing technology applets to reflect on the mathematics learning progressions.

Source: (Suh, 2016).

Teaching with technology adds another layer of complexity to the PCK framework. Understanding how to teach with technology referred to as Technological Pedagogical and Content Knowledge, TPACK, (Mishra & Koehler, 2006) integrates a third component into teachers’ specialized knowledge for teaching—integration of technology into instruction.

TPACK includes understanding how technology can be used to represent concepts, knowledge of pedagogical techniques that use technology to effectively teach content, familiarity with ways that technology can help students understand particularly difficult topics, and knowing how technology can be used to build on existing knowledge. Virtual manipulatives have been described as “interactive, web-based visual representations of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard, & Spikell, 2002, p. 373).

In our work with teachers, we ask teachers to evaluate technology applets to see how it “amplifies” the mathematics. We used the word “amplify” to mean the way in which technology offers affordances that would not be available in a physical manipulative and how the connections to the mathematics concepts are brought to clarity.

One of the practice-based activities that help teachers think deeply about the learning progressions for a specific concept is called Mathematics Tech-knowledgy across the Learning Progression, where one selects three related applets that could be used to teach and learn that concept. This helps teachers plan and teach a lesson using technology and assess a student’s understanding about a concept using a variety of representations. Figure 2.3 is a sampler of three related fraction applets that a teacher chose to use during her unit on fraction. On the right description, the teacher explains how the applet will be used to “amplify” the mathematics for a student that she was working with one-on-one.

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