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The Foundational Research Behind MathFactLab
Please consider the text below to be a work in progress.
MathFactLab was developed to provide students with a research-backed approach to online math fact fluency practice. While most math fact websites are memorization-based, MathFactLab students develop fact fluency by doing what research recommends: practicing with multiple representations and strategies - first working with the foundational facts, and subsequently using these to construct the derived facts.
The guiding principles by which MathFactLab was designed were all informed by current math educational research. Each of these principles is covered below:
There are three phases of math fact fluency development.
According to Baroody, there are three phases of math fact learning: (1) Counting strategies, in which the counting of objects is a necessary step to finding an answer; (2) Reasoning strategies, where logic, deduction, and the combination of known facts leads to the solution of an unknown fact; (3) Mastery, in which answers are produced efficiently and accurately (Baroody, 2006).
The initial phase should occur in the classroom, utilizing manipulatives under the guidance of a teacher. Developing strategies, the second of these phases, should also first be explored under the guidance of a teacher, whether it be with ten frames, number lines, rekenreks, area models, dice or other tools. Van de Walle writes, 'Phase 2 is often under-emphasized or neglected entirely, yet it is an essential bridge between inefficient counting strategies and mastery (2019).
When students are at the point where they can begin to apply strategies using manipulatives to solve fact problems more efficiently than with counting, they are ready to begin practicing with MathFactLab. At any level in the program, where a specific set of fact families are practiced, the program serves as a bridge, guiding students through Stage 2 until they achieve fluent Stage-3 mastery. Afterward, they advance to the next level in the program and repeat the process.
Memorization and drill are ineffective ways to teach math facts.
As Boaler puts it, 'Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging' (2015). Memorization skips over Stage 2 in the fact-learning progression. Students are introduced to concepts and then are expected to memorize the facts. As Van de Walle put it, 'There is strong evidence that this method simply does not work' (2019). According to Baroody, this leads to 'Inefficiency', 'Inappropriate applications', and 'Inflexibility' (2006). A memorization approach to math fact learning also inhibits the development of number sense (Boaler, 2015).
Strategy-based math fact instruction works.
There is ample evidence to show that the teaching and practicing of strategies to solve math facts is effective (Delazur, 2005; National Research Council, 2001; Cook and Dossey, 1982; Heege, 1985; Kami, 1994). Van de Walle writes, 'Research supports the use of explicit strategy instruction as effective in helping all students learn (and remember) their basic facts (e.g., Baroody, et al., 2009; Baroody, et al., 2016; Thornton, 1978; Fuson, 1992; Rathmell, 1978; Thornton & Toohey, 1984)' (2019).
Explicit strategy instruction allows students to solve facts efficiently while avoiding the pitfalls of counting or memorization. As O'Connell and San Giovanni put it, 'Teaching math fact strategies focuses attention on number sense, operations, patterns, properties, and other critical number concepts. These big ideas related to numbers provide a strong foundation for the strategic reasoning that supports mastering basic math facts' (2015).
Conceptual knowledge of the four basic operations and their relationships builds an 'organizing framework for storing arithmetic facts' (Davenport, et al., 2019). 'Children who commit the facts to memory easily are able to do so because they have constructed relationships among them and between addition and subtraction [or multiplication and division] in general, and they use these relationships as shortcuts' (Fosnot & Dolk, 2001).
MathFactLab students work with models that they are familiar with from the classroom and that build conceptual understanding: ten frames, beads, number lines, area models, dice, and bar diagrams. With these models, students practice strategies that help them construct new math fact knowledge by tapping prior knowledge. For example, students practice their near doubles (6+7, 9+8) using ten frames by adding one to the doubles facts that they already know (6+6 or 8+8). In multiplication, for example, students learn how to solve 6x4 by adding two groups, each composed of 3 fours represented with dice.
Students should first learn the foundational facts and use these to construct the derived facts.
MathFactLab is organized into stages, each broken down into multiple levels. The initial stage, 'Basic Part I' - whether in our addition/subtraction or our multiplication/division learning modes - focuses on developing fluency with the foundational facts. 'Basic Part 2' teaches strategies that use the foundational facts to build the derived facts.
The levels within each stage focus on a group of fact families that can all be practiced using the same set of strategies. The levels are ordered so that easier strategies are introduced first and can be generally applied to larger sets of fact families. This leaves fewer fact families to be learned using the more difficult strategies (Thornton, 1978; Kling & Bay-Williams, 2015). For example, Level A's simple plus-one strategy covers eleven fact families while the more challenging strategies of Level K(+7) only need to be applied to two. The order of these levels and the strategies covered at each level follow the guidance of current research (Watanabe, 2003; Van de Walle, et al., 2019; Bay-Williams & Kling, 2019; O'Connell & SanGiovanni, 2011; Davenport, et al., 2019).
Fact practice should enhance and be grounded in number sense.
Boaler defines number sense as a 'deep understanding of numbers and the ways they relate to each other.' When students have number sense, they can 'use numbers flexibly' (2015). By helping students to discover multiple routes to solving math facts, MathFactLab helps students become flexible mathematicians, leading them to current and future success in mathematics.
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