Abstract
High-leverage practices are essential components of specially designed instruction required by the Individuals with Disabilities Education Improvement Act. High-leverage practices such as explicit instruction and use of instructional and assistive technology have research evidence demonstrating their effectiveness when teaching students with learning and other disabilities. Although high-leverage practices are appropriate for many content areas, this article illustrates their use when teaching number sense. Number sense involves fluidity with (a) estimating and naming quantities, (b) evaluating accuracy of answers, (c) calculating mentally, and (d) drawing or representing problems. Development of number sense predicts later math achievement for students with and at risk for learning disabilities, making it an essential skill for teachers to address. Peppering common number sense activities with high-leverage practices enhances their impact for students with learning disabilities.
In 2017, the Collaboration for Effective Educator, Development, Accountability, and Reform (CEEDAR) Center and Council for Exceptional Children (CEC) created a series of practices that support effective instructional practices in special education (McLeskey et al., 2017). High-leverage practices (HLP) are essential components of specially designed instruction required by the Individuals with Disabilities Education Improvement Act (2004; McLeskey et al., 2017). According to McLeskey et al., the 22 HLPs “represent the essence of effective practice in special education” (p. 9) gleaned from an extensive review of research that included students with learning disabilities. Although HLPs include effective practices in assessment, collaboration, and social-emotional-behavioral areas, this article focuses on the fourth category, instruction. When using HLPs during instruction, teachers “maximize academic learning time, actively engage learners in meaningful activities, and emphasize . . . positive approaches across tiers of instructional intensity” (Riccomini et al., 2017, p. 69).
Teachers working with students with LD often plan activities and games to promote student engagement and provide repeated practice (Lock & Gurganus, 2004). Whereas the activities may not have specific research support, teachers’ use of HLPs when designing and implementing them may represent specially designed instruction. This article centers on five HLPs used to plan and implement specially designed number sense instruction for students with learning disabilities (LD): explicit instruction (HLP 16), instructional and assistive technology (HLP 19), strategies for active engagement (HLP 18), adapting curriculum and materials (HLP 13), and providing feedback (HLP 22). While each is designed to address number sense individually, teachers are encouraged to combine HLPs for even more impact on learning (Riccomini et al., 2017).
Learning Disability and Number Sense
Dyscalculia, a specific learning disability in mathematics, affects as many as 6% of the overall elementary school population (Wong et al., 2017). Students with LD in mathematics exhibit difficulties in several areas such as procedural steps in problem solving, representing and retrieving basic facts, and quantity concepts (Gersten & Chard, 1999; Wong et al., 2017). Mathematical thinking largely depends on the foundation of number sense (Bryant et al., 2016): the ability to understand what numbers mean and to quickly and accurately use those numbers to solve problems (Gersten & Chard, 1999). Number sense, infused concepts that appear throughout mathematical concepts (Lock & Gurganus, 2004), includes (a) estimating and naming quantities, (b) evaluating accuracy of answers, (c) calculating mentally, and (d) drawing or representing problems (Gersten et al., 2005). Children who struggle with number sense, including those with LD, are likely to experience difficulties with more complex mathematical concepts (Bryant et al., 2016; Gersten et al., 2005). Efficiency in counting and knowledge of basic facts are positively correlated with later math achievement for students with LD (Bryant et al., 2016). Consequently, explicit number sense instruction builds a foundation for more complex mathematical reasoning and problem solving.
Peppering common number sense activities with HLPs enhances their impact for students with LD (McLeskey et al., 2017). Explicit instruction (i.e., HLP 16; Doabler & Fien, 2013) provides the framework for incorporating other HLPs as teachers vary the intensity of instruction (Riccomini et al., 2017) to meet students’ diverse learning needs.
High Leverage Practice 16: Explicit Instruction
Explicit instruction includes “showing and telling students what to do or think while solving problems, enacting strategies, completing tasks, and classifying concepts” (Riccomini et al., 2017, p. 14) and is a framework for effective instruction (Riccomini et al., 2017). Thinking aloud to demonstrate number sense skills is one approach to explicit instruction. Teacher think alouds provide step-by-step demonstrations, examples and non-examples, and organization of knowledge (Riccomini et al., 2017 p. 25; see Figure 1). By thinking aloud while using manipulatives or drawing pictures, teachers illustrate problem-solving strategies and draw students’ attention to critical task features. Preparing for thinking aloud involves selecting an appropriate set of problems, determining critical steps in solving the problem to emphasize, and preparing support materials such as manipulatives or cue cards (Conderman & Hedin, 2011). Explicit instruction that incorporates thinking aloud allows teachers to address how to solve problems and also why to use a particular approach (e.g., “I regrouped from the tens column because . . . ”). Describing why as well as how deepens learning by revealing underlying math principles and reasoning.
Teacher think aloud with virtual or actual counters for addition. This sample teacher think aloud provides a model of how a teacher solves a problem using virtual counters in a missing addend problem.
Students’ think-alouds (see Figure 2) provide teachers with formative information on which to base scaffolded supports (HLP 15), and positive and corrective feedback (HLP 22). Students thinking aloud while using manipulatives and drawings (a) rehearse steps in the procedure, (b) evaluate their answers as they verbalize and reflect on their representations, (c) have opportunities for self-correction, and (d) use math-related vocabulary and communication skills. Word walls or desktop visuals with appropriate math vocabulary assist students during problem-solving. Having students explain their reasoning when they have the correct answers is essential, especially when students have only partially mastered a skill or procedure (HLP 21, maintenance and generalization). As students verbally restate (i.e., rehearse) problem solving steps, they have additional opportunities to practice and respond (HLP 18, active student engagement), which supports memorization. Teachers respond to student think-alouds by delivering specific feedback (HLP 22, feedback to student) such as error correction, affirmations, modeling appropriate math vocabulary use, or asking guiding questions to deepen learning.
Student think aloud with virtual or actual number line for subtraction. The think aloud provides an example of a student sharing how he solved a subtraction problem using a number line to subtract. The student makes a mistake and the teacher provides corrective feedback and modeling.
During the provided think aloud illustrations, teachers and students use manipulatives or drawings to represent and solve problems. When physical manipulatives are not available or practical, teachers can incorporate supportive technology into instruction to enhance learning.
High Leverage Practice 19: Supporting Technology
Technology is ubiquitous. When teachers incorporate instructional and assistive technology (HLP 19) to enhance and support instructional practices, students benefit (McLeskey et al., 2017) through scaffolded supports, active engagement, and reduced cognitive demands that technology provides (Shin et al., 2017). Technology enhances but does not supplant instruction; therefore, teachers must be intentional in its use. Students still need modeling, ample guided practice, and opportunities to engage in independent practice. One way to integrate technology into instruction is by using virtual manipulatives to support math instruction.
Virtual Manipulatives
Using visual models to represent and solve math problems supports the development of number sense (Shin et al., 2017). Virtual manipulatives, in place of concrete objects or manipulatives, are one way to do this (Lock & Gurganus, 2004; see Table 1). Many virtual manipulatives are free and compatible with classroom technology already available to teachers. Sites like those listed in Table 1 (e.g., Toy Theater; Gaspard, 2020) provides a variety of lesson plans and guidance in the use of manipulatives to familiarize teachers with tools before exploring with students.
Websites With Virtual Manipulatives and Math Activities Related to Number Sense.
Note. This table provides links to video demonstrations and virtual manipulatives and other mathematics supports for building number sense.
Similarly, The Techie Teacher (Smith, 2020) website has links to a range of virtual manipulatives for use with students through middle school. The Smart Exchange (2021) website offers a large number of ready-made activities that either model how to represent problems or allow students to represent problems.
One type of visual model that teachers can use to develop student number sense is a 10-frame, typically constructed as an array consisting of a 2 by 5 display of boxes stacked upon each other. Teachers place counters within the boxes to visually represent numbers up to 10, laying the foundation for basic computation and mental math (Gersten & Chard, 1999). virtual number frames, similar to 10-frames, from the Math Learning Center assist students in seeing quantities as parts to a whole or equal and unequal (McGuire et al., 2012). As students increase automaticity with instantly seeing amounts, they can begin to compare quantities, becoming more efficient at counting and computing (Clements, 1999). Virtual manipulatives provide numerous opportunities for practice, responding, and active engagement (HLP 18).
High Leverage Practice 18: Active Engagement
Another HLP critical to achievement for students with LD is active engagement (McLeskey et al., 2017). Techniques such as peer tutoring, choral response, and repeated practice provide students with purposeful, interesting, and motivating ways to engage with learning tasks. Klem and Connell (2004) found that students’ active engagement predicted academic and behavioral outcomes. Teachers who promote active engagement (a) foster a positive learning environment, (b) build on student strengths and interests, (c) convey to students why knowledge and skills are important, and (d) maintain high success rates to build students’ self-efficacy (Marzano & Pickering, 2011). To promote active engagement in number sense activities, teachers can use games that allow students high rates of participation in a fun format, such as Magic Numbers, a mental calculation game.
Magic Numbers
During Magic Numbers, all students complete mental calculations individually using numbers suggested by the teacher. The students individually think of a number, then the teacher prompts a chain of operations (e.g., add 2, subtract 5; see Table 2). In the last step, students subtract their original number, meaning that the teacher cancels out students’ original numbers (any number subtracted from itself is zero). Therefore, all students end with the same number—the sum/difference of the chain prompted by the teacher. During Magic Numbers, students remain actively engaged in mental calculations, continuously adding and subtracting in their heads.
Magic Number Script.
Note. This script is an example of the Magic Numbers game, including addition and subtraction within 20. The bold 3 is the number that the student selected at the beginning of the game. Students may use finger counting to check their mental math. Identity property of addition: Zero added to any number is the number itself; any number subtracted from itself is zero.
Teachers adjust the level of difficulty of Magic Numbers to individuals’ learning needs by changing (a) the range of numbers added and subtracted (e.g., from 1 to 10 vs. 1 to 50), (b) the number of steps in the chain of operations, (c) the complexity of the mental computation (e.g., does or does not require regrouping, subtracting to result in negative numbers), and (d) students’ think time (e.g., 10 vs. 20 s). Teachers base these decisions on students’ skill levels and learning targets; that is, adapting curriculum and materials for instruction (HLP 13).
High Leverage Practice 13: Adapting Curriculum and Materials
Adapting curriculum and materials (HLP 13) creates access to learning for students with LD. Planning for adaptations, teachers identify barriers to learning experienced by particular students, review grade-level standards, and pinpoint content enhancements (McLeskey et al., 2017) that will assist learners. The previous number sense HLP examples have outlined some ways to adapt activities to meet the specific strengths and needs of learners (e.g., Magic Numbers, supportive technology). The final activity, Find It! (Fullerton, 2015) shows ways to differentiate a subitizing activity, another component of number sense (Clements, 1999).
Find It!
Children’s ability to quickly and accurately estimate how many is a component of number sense. Recognition of quantities based on visual patterns, or subitizing (Clements, 1999; Conderman et al., 2014), facilitates efficient estimation. Automatically recognizing dot patterns on a die and saying quantities without counting are examples of subitizing. Find It!, a game based on visual pattern recognition, builds number sense using cards with a dot pattern or a numeral in each cell (see Figure 3).
Dot pattern bingo board is inspired by https://buildmathminds.com/freebies/ (used with permission). Students roll a die and cover the dot pattern on their card that matches the number on the die to support subitizing. Mix of numerals and dot patterns promote number recognition and subitizing. Boards can also be constructed for play with two dice.
The game is played following the same rules as bingo. Players roll a die or dice, locate a cell with the same quantity on the Find It! card, and mark the cell with an X or a counter. Teachers can teach the game using smart technology or a projector during explicit, whole-group instruction before students play individually with cards that are adapted to their proficiency levels. Teachers can quickly adapt Find It! to the diverse learning needs of students with LD.
Teachers can differentiate Find It! cards for diverse learning needs by changing the number of dots in each cell, the configuration of the dots, and mixing dots and numerals (see Figure 3). For example, very young children counting to six may need to use only one die and a Find It! card with patterns identical to those on the die. As students’ ability to subitize improves, teachers provide cards with dot patterns that differ substantially from those on the die or include dot patterns up to 12 (i.e., use two dice or dominoes). In these ways teachers adapt the materials, individualizing number sense activities. However, providing Find It!, even with adapted playing cards, does not eliminate the need for explicit instruction or feedback to students as they participate in the activity.
High Leverage Practice 22: Positive and Constructive Feedback
Although McLeskey et al. (2017) described HLP 22 as a distinct practice, teacher feedback should be woven throughout explicit instruction. Teachers who incorporate positive and constructive feedback (HLP 22; McLeskey et al., 2017) guarantee that students practice appropriate skills. When students with LD receive rich feedback on task performance, they reconfigure and consolidate their thinking. Students with LD who are learning number sense require specific feedback that includes what they did correctly. Consider the following fictionalized scenario sequences: Cho, when you were thinking aloud, I heard you use some good math words, like subtract and minus. That shows me you have been listening when I modeled an example.
Conversely, students need specific feedback on steps they either struggled to grasp or completely glossed over. For example, when Cho makes a “hopping” error with numberline subtraction (see Figure 2), his teacher responds by saying: My turn, watch me. I put my finger on 14, then I hop backward. I start counting AFTER my first hop. [Teacher points, counts: five, four, three, two, one]. Watch one more time. When do I start counting? Yes, after my first hop. [Teacher points, counts: five, four, three, two, one], and I land on nine. So 14 minus 5 equals 9.
This explicit feedback supports the acquisition of knowledge and skills and deepens learning. Therefore, feedback should be peppered throughout specially designed instruction (Riccomini et al., 2017).
Conclusion
Fostering number sense has the potential to impact positively later mathematics reasoning for students with LD. However, games and activities often used to build number sense must be paired with specially designed instruction that includes HLPs such as explicit instruction and corrective and positive feedback. It is imperative that teachers of students with LD purposefully engage students in authentic experiences that build number sense. The number sense knowledge and skill activities described here derive their power from the HLPs used in their delivery and the ways the teachers adapted activities to students’ varying skill levels and learning targets. Teachers can incorporate explicit instruction, technology, active engagement strategies, and rich feedback as a means of creating specially designed instruction for their students with LD.
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) received no financial support for the research, authorship, and/or publication of this article.