Abstract
As American students struggle with basic mathematical skills, the importance of math fact fluency has gained the attention of educators and researchers. Generalization of math fact fluency is also important for the transfer of skills to other settings and formats, assisting students in the completion of more varied and complicated math tasks. This column explores technology in the classroom and use of mobile applications (i.e., apps) to train students in fluency and generalization of math facts. Key features of apps that teachers should consider for classroom use are described and ways in which apps can provide students with a variety of stimulus and response conditions to promote fluency and generalization of math facts are highlighted.
Results from the most recent Trends in International Mathematics and Science Study (TIMSS; Mullis, Martin, Foy, & Arora, 2012), show the United States ranked 24th among 57 countries. A consistent and key finding across recent reports from the National Center for Education Statistics (NCES; 2011) and the National Mathematics Advisory Panel (NMAP; 2008) is that American students are struggling with basic mathematics skills. In light of these findings, researchers and educators have focused their efforts on effective and efficient remediation strategies with the goal of increasing students’ fluency with basic computation (Poncy, Skinner, & Axtell, 2010). The column provides a brief overview of the importance of fluency and generalization training with basic math facts along with a discussion of how mobile applications, commonly referred to as apps, can be used for training students in fluency and generalization of math facts.
Training for Fluency and Generalization
The importance of fluency with basic math facts in completing more advanced tasks related to mathematics and computing is clear. Fluency refers to the rapid and accurate responding to a group of stimuli (Parkhurst et al., 2010). In the case of math facts, fluency is being able to respond to basic math facts in the four operations (i.e., addition, subtraction, multiplication, and division). Commonly cited benefits of fluency include lower levels of math-related anxiety and less demand on working memory when completing more complex tasks (Parkhurst et al., 2010; Poncy, McCallum, & Schmidt, 2010). Students with disabilities who have difficulty with math in the early grades tend to become disenfranchised with science, technology, engineering, and mathematics (STEM) content as early as middle school (Marino, 2010). Therefore, gaining fluency with basic math facts becomes an important focus in the early elementary grades.
Several interventions have shown positive effects on math fact fluency, including taped problems (TP; McCallum, Skinner & Hutchins, 2004); cover, copy, compare (CCC; Skinner, Turco, Beatty, & Rasavage, 1989); and detect, practice, repair (DPR; Poncy, Fontenelle, & Skinner, 2013). In the TP intervention, students are given problem lists and attempt to answer each problem before an audiotape provides the correct answer. Varying time delays are used to encourage quick, accurate answers with the goal of increasing automaticity. The CCC intervention requires students to follow a set of procedures in which they cover a problem, write the problem and answer, and then uncover and evaluate the accuracy of their answer (Poncy, Skinner, & McCallum, 2012). The procedure is repeated for a predetermined length of time. The DPR intervention adds a test-teach-test component to the CCC procedures (Poncy et al., 2013). In the detect stage, the student completes a paced pretest to identify facts for practice. During the practice stage, the student uses the individualized problem set for review using the CCC procedures. Finally, an alternate version of the detect test is given in the repair stage, whereby practice problems are provided within a larger pool of problems to evaluate generalization under a timed condition. Common elements across these interventions include multiple response opportunities, immediate feedback, and error corrections to effectively increase rate and accuracy of response (Poncy, Skinner, & Jaspers, 2007).
The question of what happens once students reach fluency with math facts has received limited attention. According to Haring and Eaton’s (1978) instructional hierarchy, a natural progression of skill training once students reach fluency with a skill is to train students to perform the skill under untrained and novel conditions (i.e., generalization). Generalization can occur along three dimensions: stimulus, response, and time (Cooper, Heron, & Heward, 2007). In the context of learning basic facts, for example, stimulus generalization is said to have occurred when a student trained to complete math facts presented in vertical format is able to transfer the skill to math facts presented in horizontal format. Response generalization occurs when a student responds to math facts by entering the answer on a tablet touch screen during instructional sessions and later is able to respond to the same facts by writing the correct answer on an exam using a pencil. In the third dimension, also known as maintenance, a student is able to maintain the skill and respond to math facts accurately over a period of time once training in the skill has been terminated. In classroom learning, these three types of generalization are not mutually exclusive but occur simultaneously. For example, students who use flashcards to memorize addition facts (e.g., 5 + 3 = 8) in the instructional setting complete a similar operation after a few months (i.e., maintenance) when presented in the form of a word problem (i.e., stimulus generalization) by writing the correct answer (e.g., response generalization).
Despite the existence of a technology of generalization wherein specific strategies have been discussed to plan and promote generalization and maintenance of skills (Cooper et al., 2007; Stokes & Baer, 1977), there are only a handful of studies that report the effects of fluency training on generalized outcomes. For example, Miller, Skinner, Gibby, Galyon, and Meadows-Allen (2011) found evidence of generalization when they assessed multiplication facts in inverse form (e.g., 7 × 5) when the reverse (i.e., 5 × 7) was practiced during intervention. Codding, Archer, and Connell (2010) reported evidence of generalization to fractions and word problems with training in multiplication facts. However, Poncy, Duhon, Lee, and Key (2010) failed to see generalization from one math operation to another (i.e., addition to subtraction) when the same digits were presented in a different order.
Some students, especially students with disabilities, can have difficulty using their newly learned skills in settings that differ from the original setting or context in which the skill was first learned (Heward, 2013). In other words, there is a failure to transfer skills from instructional setting to generalization setting especially when there are changes in the stimulus presentation and/or response requirements of the skill. Failure to generalize will result in retraining the student, a process that can be both time- and resource-intensive for teachers. Instead of relying on a “train-and-hope” method to see generalization, researchers urge educators to use built-in instructional approaches to promote generality and maintenance of skills (Codding & Poncy, 2010; Stokes & Baer, 1977). It is important that students are exposed to a variety of stimulus conditions and response requirements in the instructional process (Cooper et al., 2007; Daly, Martens, Barnett, Witt, & Olson, 2007). With advancements in the digital world, the use of classroom technology to program for both fluency and generalization is a true possibility.
Using Technology in the Classroom
The use of technology in educational settings goes by many names: computer-assisted instruction (CAI), information and communications technology (ICT), education technology, and instructional technology are just a few. The word technology has been used to refer to various tools used in educational settings, including textbooks, writing instruments, radio, film projectors, digital media, interactive whiteboards, and computers (Buckingham, 2008; Cuban, 1986, 2001). In the recent two decades, what we mean by technology in education has increasingly come to mean “software” (Lynch, 2013a). The term technology is used in reference to desktop computers, laptops, tablets, and any handheld mobile devices that run software applications available commercially for purchase or download. Funding and interest in the use of tablets and mobile apps, like those available through the iPad, has grown in recent years (Lynch, 2013b). While the number of software-powered technologies made available to educators grows every day, researchers collectively caution schools to prioritize pedagogical ends (Castek & Beach, 2013), not merely reinforce the status quo through the use of technology (Ready, Meier, Horton, Mineo, & Pike, 2013).
One of the reasons it is so difficult to study the use of technology in schools is because of the myriad variables that influence the implementation of software-powered technologies in school settings (Dede, Honan, & Peters, 2005; Sandholtz & Reilly, 2004) as well as the rapid pace of new software development. In addition to the social context that a school setting provides, the integrity of technology implementation is challenging to ensure due to the lack of thorough planning and support structures in place to help teachers and students overcome inevitable snags (Meier, 2005). When software-powered technologies are strategically used to further one’s pedagogical ends, the results can be promising.
Teachers can choose from a wide array of apps available to facilitate fluency with math facts. A recent analysis of the education category of Apple’s App Store revealed that 50% of top-selling (top 25) apps target elementary-age children, with math ranking as the second most popular subject area (Shuler, 2012). Many of the math fact apps share similar pedagogical features, some more learner-friendly than others. A side-by-side comparison of the pedagogical features of three specific apps is presented in Table 1.
Side-by-Side Comparison of the Pedagogical Features of Three Math Applications.
As part of a larger research study, we used the Math Drills app in a third-grade integrated cotaught classroom where 22 students with and without disabilities received instruction in the general education classroom, staffed with one general education and one special education teacher.
In addition to performance outcome data based on students’ fluency, we collected multimodal data while students used the app; specifically, we video-recorded the students’ activity and transcribed it for multimodal elements, like students’ posture, gestures when interacting with the device, and hand motions. The app has several features that not only can prove advantageous for just fluency building but can also be used to program for generalization. Using the Math Drills app as an example, there are ways in which an app can be customized to provide students with a variety in stimulus conditions and response requirements, in turn promoting generality of math facts.
Using Apps for Generalization Training
One of many available tactics to promote generalized outcomes involves teaching the student to respond to different stimulus conditions and response examples and then probing for generalization to untaught stimulus examples (Cooper et al., 2007). Also known as teaching sufficient examples, this tactic operates on the logic that students will be more likely to respond to untrained examples or situations only when they are exposed to a wide range of examples during instruction.
Themes
The Math Drills app provides users with the option to choose from six different themes. The themes differ in the color and style of font used to present the numbers, including the background. Once students gain fluency with one theme, more than one theme can be selected to appear in random order during drills. The color options used to show correct and incorrect responses include red and green, yellow and blue, and light and dark. This feature can be particularly helpful for students with color blindness who may have difficulty seeing red and green.
Arrangement
Users have the option of selecting the vertical-only format, the horizontal-only format, or a combination of both formats. Teachers can start students off with one format, and as they build fluency, the other format can be introduced before moving to drills where problems in both formats appear in random order. The app allows users to select whether the keypad should be located on the bottom-right corner for right-handed users or bottom-left corner of the screen for left-handed users. Toggling with an “on/off” switch on the screen will allow the display on an extralarge keypad when the device is used in portrait mode.
Assistants
When practicing the drills in “review” mode, the user can choose to enable the “assistants” mode that in turn activates a number line with an animation of arrows counting along the number line. For example, for “2 × 5,” there is an arrow that skip-counts from 0 to 10. The duration of the animation interval can be adjusted starting from 1 to 30 s. Students who need visual supports to help memorize math facts can benefit from this feature. These supports can be gradually faded as students gain fluency with the math facts.
Answers
The app provides three response options. Users can: (a) enter the “numbers” in response to the math problem (e.g., 5 + 4 = ), (b) enter the “operators” to complete the math problem (e.g., 5 4 = 9), or (c) enter the “comparators” to complete the math problem (e.g., <, =, or >). An “on/off” feature allows users to include a placeholder with a question mark, symbolizing which of the three features needs attention.
Customize Math Facts
Starting with times tables for smaller numbers, like 2, 3, and 5, before moving to other numbers is a common practice. Teachers can customize math facts so that students can gain fluency with smaller numbers (e.g., 2 × 9) before introducing practice with the inverse fact (e.g., 9 × 2). Using curriculum-based measurements in which students are assessed on their math facts using 1-min timing, the teachers can customize the app with specific facts students are yet to master.
Response Input
The arrangement of numbers (e.g., 0–9) on a computer keyboard or calculator device differs from the way the numbers are arranged on a telephone keypad. For example, the numbers 1, 2, and 3 form the first row on a telephone keypad, whereas the same three numbers form the third row on a computer keyboard or calculator device. Students can be trained to use one type of keypad before being introduced to the second type. This way, students will have practice in both types of keypad. The Math Drills app also provides users with options to either use one of the keypads to enter the responses or select the correct answer when given three choices (i.e., multiple-choice format). Varying the response requirements during practice drills will better prepare students to perform the same tasks in typical test-taking conditions.
Error Correction
Immediate feedback and error correction is an important element of effective instruction. Using an “on/off” feature, users can set the app to correct the answers immediately before moving on to the next problem or correct the answers at the end of the drill set. In either case, the students receive immediate feedback on correct and incorrect responses with the opportunity to try again until they get it correct. As shown in Table 1, the feedback feature is common across all three apps.
Data Collection
The app tracks students’ performance on tests by recording the accuracy and rate of response. Using a racing theme, the scores are reported as the “duration” it takes to complete the test, and the number of incorrect responses is indicated as “pit stops.” Teachers can track students progress within the app.
Implications for Classroom Practice
Classroom teachers should consider the affordances of apps in the context of their own classrooms. Although the Math Drills app had several noteworthy features, there were some multimodal elements we observed as potentially affecting students’ use of the app. First observed were thematic incongruities. Thematic incongruities refers to ways in which the app developer presents users with multimodal texts and logics that do not align with the primary purpose of the app. For example, while the background of the app shows a traditional chalkboard, students do not write on the board’s surface, but rather, they tap on a keypad, which in turn types on the chalkboard. The subtlety of the incongruity between these two elements—the chalkboard and keypad—might distract some students even if they cannot articulate this difference.
As another example, the app attempts to make it enjoyable for students to track their own progress by providing a gamelike function in the form of race car competitions. Before students begin their drills, the app shows a traffic light, and students begin when the light turns from red to amber to green. Further, students hear sound effects associated with race cars throughout. While the idea of speed racing might be fun for some students, it does not match with the primary function of the app or other thematic qualities that situate the student’s learning in a digital classroom. Next observed were corporeal controls. Students grappled with the physicality of the device in ways that merit further attention. Two students who performed well on the app quickly folded the device cover in a manner that allowed them to place the device on the desk in front of them. They then hunched over and focused on the drills. A third student frequently flipped and waved the device as he tried to complete the drills, never gaining comfort with the device’s weight or placement. It is worth considering the extent to which students compensate for the abstract and digital nature of using apps on tablets by controlling (or not) the physical qualities in front of them: the device placement, their posture, and their hands.
Finally, analog tensions were noted. Analog tensions refer to the way students get frustrated when using the app or device by expectations based on their use of nondigital learning resources. For example, when using flash cards, students are presented with the correct answer to problems. In the Math Drills app, this was not the case. To be clear, it is not a technical limitation but rather a choice by the software developers to withhold the correct answers from students. While it would be fruitful to explore how these kinds of seemingly minor decisions by software developers support or contradict research about student learning, what is important to note is that students’ expectations, skills, and confidence to engage in some digital learning experiences are inherited from their analog experiences.
Classroom teachers should take these observations into account when using similar apps with students. While some features of such apps might help meet the needs of students with learning disabilities—the highly visual nature of the app, the use of touch interfaces that are less reliant on fine-motor skills, the use of sounds to reinforce progress and achievement—other features might be less supportive. Specifically, the physicality of the device and the lack of opportunity for students to process their solving of math problems might make similar apps better suited as an enrichment or supplemental tool to high-quality, evidence-based classroom instruction. Given the Common Core Mathematics Standards emphasis on fewer key concepts and openness to students demonstrating an array of strategies to arrive at correct answers, math apps might be viewed as serving an important but limited role in the classroom that supports students in mastering basic knowledge needed to engage in more complex problems. In our observations, math apps can be a useful tool for students across grade levels. Decisions on the frequency and duration of using the apps as supplemental instruction should be made based on progress-monitoring data and learning goals of the students. When selecting apps to use, teachers must consider not only the appropriateness of the content covered in the app but also the thematic design, some elements of which seem aimed at younger audiences and the cartoonlike style of which might turn off older learners.
Conclusion
When used skillfully and systematically, technology can be of valuable assistance to teachers in the classroom to improve students’ fluency and generalization of basic academic skills. Teachers should take the time to train students in the use of the device as well as provide explicit instruction on how to navigate through a particular app. Exposing students to a wide range of stimulus conditions and response requirements during training can prevent retraining students who fail to generalize skills to novel settings or situations. Selection of the math fact apps and the pedagogical features they possess should be carefully matched with students’ learning needs. It should be noted that these classroom technologies should be used as a supplement to sound, evidence-based teaching practice and not replace teachers in the classroom. In essence, this means using technology for what technology is good at. These kinds of uses of technology must not be confused for the kinds of radical innovations or solutions to educational challenges, as they often are. They are cognitively low level and pedagogically unimaginative when used in isolation of inquiry-based and evidence-based teaching models.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Provost’s Grant for the Summer Thinkfinity Initiative at Pace University, New York.