Using Number Talks to Support Students With High-Incidence Disabilities in Mathematics

Abstract

Number talks are increasingly used in general education mathematics classes to engage students. Yet despite the potential benefits, number talks are given limited attention for students with high-incidence disabilities in special education settings. This article presents special education teachers with both the why and, more important, the how for implementing number talks to support students with high-incidence disabilities in special education settings. Specifically, the authors address how number talks can serve as both a formative assessment and an intervention for fluency and activating students’ background knowledge to be successful in general education settings. The article also provides suggestions for implementing number talks with fidelity and flexibility (e.g., use of manipulatives, pictorial representations, and teacher explicit instruction of numerical strategies).

Keywords

intervention academic instruction mathematics skills at-risk students strategies

Number talks are increasingly used in general education mathematics classroom as a means to engage students in discussion regarding mathematical ideas of number and operation (Boaler et al., 2015; Parrish, 2011). Number talks are short activities in which the focus is on how students approach and work through a problem rather than the answer (Sun et al., 2018). It is important to note that a number talk is not just seat work or an activity to fill students’ time as the class gets settled or students enter the room. The mathematics of a number talk is also not the content of the day’s lesson but rather mathematical content students already received instruction on and were expected to have learned (Boaler et al., 2015; Parrish, 2011). Number talks are meant to deepen and expand students’ understanding of the mathematics and to build flexibility with number and operations.

The use of number talks is aligned with the Common Core State Standards for Mathematics (CCSSM), in which there is a focus on students being able to verbally communicate about mathematics and engage in discussions regarding mathematical reasoning (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010). Furthermore, number talks are meant to support mental computation, which is part of the CCSSM and an important component for developing number sense (National Mathematics Advisory Panel, 2008; National Research Council, 2001). Finally, number talks promote elements noted to support mathematical understanding (i.e., seeking and using structure, making connections, and productive struggles), which are areas noted as lacking in mathematics teaching (Hiebert et al., 2003).

As typically done with a number talk, a teacher gathers students in an area of the room or gains their attention. The teacher first establishes, or reminds students of, expectations and norms for the number talk, such as the use of hand symbols to communicate with the teacher regarding their solution and the number of strategies a student found for solving and are prepared to share with the whole class (Boaler et al., 2015). For example, a thumb up in front of one’s chest might mean the student has a strategy while holding up three fingers usually means the student has three strategies or ways of approaching the problem (Sun et al., 2018). The teacher presents the number talk, often both verbally and in writing on the board, and asks students to independently engage (i.e., quietly thinking about how to mentally solve the problem). After the teacher has given students time to think and the majority of the students have signaled they have at least one strategy to share, the teacher asks different students to share solutions, which the teachers record on the board without judging whether the given solution is right or wrong. The teacher then asks a student to explain how they arrived at their solution. During this time of sharing, the teacher’s main job is recording student strategies and asking questions to prompt students to articulate and justify what they did mathematically and why, which usually results in students using a significant amount of verbal communication and mathematical language (Boaler et al., 2015; Parrish, 2011). After one student finishes, the teacher asks for different approaches and strategies, calling on three or more students to share their ideas.

During number talk time, students are given time to think and a space to verbalize how they approached and worked through the mathematics. With a number talk, the focus exists mainly on mental math and fluency, which are demonstrated through the mathematical conversations (Humphreys & Parker, 2015). With all number talks, fluency is defined as students’ ability to use number sense and relationships and compose and decompose numbers, rather than speed and accuracy in solving problems (Boaler et al., 2015; Parrish, 2011). It is through the frequent use of number talks that teachers help students develop deeper understanding of numbers and the relationship between numbers, so as to become more flexible and efficient in their mathematical thinking and problem-solving (Parrish, 2011).

Research on Number Talks

The limited research on number talks provides support for this instructional approach and strategies for implementation at the elementary and secondary levels. For example, Okamoto (2015) found the average scores for students in a sixth-grade class on both assessments of number sense and equivalent expressions improved following a 6-week intervention of number talks. During this time period, the students engaged in number talks 2 to 3 times per week for 15-min increments. Newell and Orton (2018) shared how teachers can implement number talks across a variety of elementary grade levels. They suggested number talks that present images can launch discourse of multiple mathematical ideas among students. Gerstenschlager and Strayer (2019) provided strategies for implementing what they coined “statistical number talks” in middle school mathematics classroom (p. 363) and suggested number talks allow for teachers to evaluate prior knowledge as well as serve as an option for formative assessment.

Students With High-Incidence Disabilities and Mathematics

Students with high-incidence disabilities (i.e., students with learning disabilities, emotional-behavior disorders) struggle with mathematics. Researchers suggest approximately 5% of school-age children experience a mathematics disability, and other students with disabilities without an identified mathematical disability still experience difficulty or receive support in mathematics (Soares et al., 2018). In contrast to the general education mathematics literature and educational psychology literature, limited attention exists on strategy instruction in mathematics for students with disabilities (Zhang et al., 2016).

The existing research suggests some instructional strategy approaches targeting mathematics (e.g., visual representations; concrete–representational–abstract [CRA]) are evidence-based (Bouck et al., 2018; Jitendra et al., 2016). Researchers and practitioners work extensively to teach students with high-incidence disabilities to be able to solve mathematical problems using only numerical strategies (i.e., a focus on the numerical representation; Siegler et al., 2011). In particular graduated sequence of instruction models (e.g., CRA, in which students transition from solving mathematical problems first with concrete manipulatives, then with representations [or drawings], and then abstractly with numerical strategies), this is referred to as the abstract phase in which students solve mathematical problems without visual supports (i.e., concrete manipulatives or pictorial drawings) and use numbers (Hinton & Flores, 2019). Numerical strategies are important for students with disabilities as they are more efficient (Zhang et al., 2016).

Number Talks and Students With High-Incidence Disabilities

While number talks are gaining in popularity in general education mathematics settings at the elementary and secondary levels, the traction and support in special education mathematics settings are more limited. In other words, limited attention exists on number talks to support students with high-incidence disabilities struggling in mathematics. Yet, number talks are appropriate in settings that support students with high-incidence disabilities who struggle in mathematics, including special education settings and delivered by special education teachers. This article discusses why and how special education teachers can implement number talks for students with high-incidence disabilities.

Why Implement Number Talks

A special education teacher who provides intervention and supports students with high-incidence disabilities can use number talks to serve multiple purposes, such as an intervention to address and develop fluency, as a formative assessment of mathematical knowledge or skills, and as preparation for mathematical experiences in general education settings. As an intervention, number talks provide special educators a means of addressing student fluency, when fluency is defined as students’ ability to solve computational problems by understanding numbers and being able to compose and decompose numbers to address this work, in contrast to just speed and accuracy (Boaler et al., 2015; Parrish, 2011).

The use of a number talk to support fluency provides special educators with alternatives to such methods as cover–copy–compare (CCC), in which students look at a problem and its answer, cover both, rewrite them, and compare their answer with the original (Riccomini et al., 2017) as well as the use of flashcards, drill sheets or apps, and timed tests, which all focus on memorization. For example, rather than asking students to complete multiplication worksheets or multiplication flashcards, teachers can engage in number talks focused on the numbers, the relationship between the numbers, and the operation of multiplication itself. With multiplication-focused number talks (e.g., Devin baked cookies. He had enough cookie dough to fill 3 pans and each pan had 12 cookies on it. How many cookies did Devin bake?), a special education teacher can have students present and discuss a variety of strategies to solve such problems, such as skip counting (i.e., in this case by 3s), drawing a picture, repeated addition (e.g., 12 + 12 + 12), and partial products (e.g., 3 × 10 + 3 × 2). Use of the number talk not only involves students finding a solution but also connections between mathematical ideas, as well as the manipulation of numbers in a way that makes sense to the students, something successful mathematics students do frequently (Boaler et al., 2015). Both elementary and secondary students benefit from a focus on reasoning and strategies, as opposed to just finding the answer. They also benefit from engaging in conversations that require articulating and justifying ways to solve a problem as well as having to make sense of and analyzing the justification of classmates (Sun et al., 2018).

Number talks can serve as a means of formative assessment, offering special education teachers insight into students’ understanding, thinking, and use of numbers during individual student’s verbal explanation of how they solved problems. These talks can serve as an important addition beyond curriculum-based measurements to determine whether students are responding to interventions and whether they need additional interventions with regard to particular mathematics concepts. For example, a student who is unable to articulate and explain which is bigger, 57 or 4 tens and 16 ones, when given as a number talk problem, is likely struggling with place value. The special education teacher can then target place value as a mathematical concept to address.

Finally, special education teachers can implement number talks to prepare students with high-incidence disabilities for mathematical experiences in general education settings. By engaging in this practice, teachers are preparing students with disabilities to actively participate in number talks in general education settings and using number talks to activate students’ background knowledge to more actively engage in the general education content lesson. As general education teachers increasingly use number talks to build fluency and activate students’ background knowledge, students with disabilities need to feel comfortable to engage in number talks with their peers. Through engaging in number talks in a smaller and more supported environment, special education teachers can help students with disabilities to practice participating and verbally sharing their thinking prior to doing so in a larger general education environment. By special education teachers presenting similar number talks to what students may experience later in their general education class, they allow students the opportunity to make sense of the practice and get used to the norms and expectations while teachers can simultaneously assess prior skills and/or knowledge as well as activate students’ said prior skills and knowledge. For example, near the end of fifth grade and early in sixth grade, many general education mathematics curricula present fraction operations. To prepare students for this unit of study, a special education teacher might select fractions for their number talk with tasks focused on the meaning of fractions as well as ordering and comparing fractions in an effort to connect and build upon precursor skills and review foundational understanding of fractions. As an example, a special education teacher might present the following number talk to their students prior to the work with fractions in the general education settings: “I am going to put a fraction on the board. You are to decide if it is greater than or less than $1 / 2$ , and be ready to explain how you know; and then writes 3/8 on the board.”

How to Implement Number Talks

When special education teachers implement number talks, the first task is to establish norms and procedures. Although the class size may be smaller in a special education setting than a general education setting, teachers need to develop and maintain similar expectations for a number talk. These norms include (a) the teacher bringing students to a central area, which in a special education classroom may be a rug, table, corner, or simply attention to a particular place (e.g., whiteboard); (b) the teacher explaining that during a number talk, all students are to think and work independently while trying to solve the problem presented; (c) students indicating whether they have one or more strategies for finding the solution by displaying one or more fingers in front of them; (d) the teacher asking for any and all solutions students have, being sure not to indicate whether they are correct or not; (e) the teacher asking students to explain how they arrived at their solution, making it clear that students are to listen to each other’s strategies and think through what is being presented; and (f) before ending a number talk, a teacher should clarify any misconceptions and address inaccuracies in strategies as well as the solution. In other words, a special education teacher should not end a number talk without responding to the whole class, through questions, or talk as needed, regarding any incorrect solutions or strategies shared.

The next task for special education teachers when implementing a number talk is to determine the line between fidelity and flexibility for students with disabilities. The basic inherent features of the number talk should be maintained when special educators implement talks, such as students discussing their reasoning and strategies to approaching and working through mathematical problems. However, flexibility should also exist. Flexibility may include such moves as the infusion of manipulatives, visual representations, the teacher modeling and sharing strategies for how to work with the numbers and operations involved in the given problem, and other appropriate accommodations. Although these elements are not standard in traditional number talks, when implementing number talks for students with high-incidence disabilities, additional facets must be considered to facilitate participation, access, and achievement.

Special education teachers may need to provide students with appropriate manipulatives when engaging in number talks, despite this not being standard or encouraged with number talks in general. Special education teachers, in accordance to research, should select manipulatives that are considered bland (i.e., less realistic), such as base 10 blocks, algebra tiles, faction tiles, as opposed to more realistic ones (i.e., called rich), such as money or buttons (Carbonneau et al., 2013; Peltier et al., 2019). Teachers should experiment as to when and how long they might incorporate manipulatives during number talks. Educators may find they need to start with manipulatives when addressing a newly identified mathematical gap concept, and as more number talks are implemented, the use of manipulatives can be faded and eventually discontinued. It is not recommended that manipulatives be used throughout the whole year for students with high-incidence disabilities as the goal is to work to improve students’ numerical mental strategies.

Although number talks focus on mental computation and understanding number and number relationships by composing and decomposing numbers, when students with high-incidence disabilities struggle, special education teachers can support student understanding of number or number relationships and the skills related to composing and decomposing numbers through visual representations (Jitendra et al., 2016). Given visual representations, such as pictorial drawings, are an evidence-based practice for students with disabilities (Jitendra et al., 2016), encouraging students to use such strategies when they first engage in number talks or with newer mathematics content in number talks can support student participation and understanding of the mathematics. Similarly, as with manipulatives, educators can start with visual representations and then fade them over time, with the end goal of students applying numerical strategies to solving the problems posed in a number talk.

As students with high-incidence disabilities may initially lack vocabulary, numerical strategies, and an understanding of what it means to talk about how one manipulates numbers when asked to share their thinking during number talks, another flexibility to the implementation of number talks in special education classrooms involves the special education teacher modeling such strategies. While a teacher should work to clarify any misconceptions or inaccuracies, special education teachers may also actively share strategies students do not yet know or think of during the number talk. For example, if a teacher presented the number talk, “Devin baked cookies. He had enough cookie dough to fill 3 pans and each pan had 12 cookies on it. How many cookies did Devin bake?” and students shared a variety of strategies (e.g., skip counting, repeated addition), but not the idea of partial products, the teacher should share this effective and efficient strategy with students to expose them to such strategies. Likewise, with the number talk, “Olivia had 19 football cards. She just got a new pack of football cards with 18 cards in the pack. How many football cards does Olivia have now?” if students do not provide partial sums via place value as a strategy, the teacher should explain it (e.g., breaking 19 and 18 both into 10 plus a number, such as 10 + 9 for 19 and 10 + 8 for 18. 10 + 10 is 20 and 9 + 8 is 17; 20 + 17 = 37).

Fluency

To support student fluency through number talks, teachers need to make a concerted effort to explicitly highlight and present how numerical strategies involve flexibly composing and decomposing numbers when shared by students or teachers, as previously noted. In the number talk example involving double-digit addition with Olivia and her football cards, if no student shares the idea, the teacher would explicitly explain how one could break 19 into 10 and 9 and that 19 means 1 ten and 9 ones. Likewise, explicitly explaining how 18 can be decomposed into 10 and 8, or in other words, 1 ten and 8 ones. The teacher can then model, through a think-aloud, about how to then add by chunks or place value (e.g., 10 + 10 = 20 and 9 + 8 = 17; 20 + 17 = 37). By implementing number talks regularly and routinely (i.e., daily or multiples times per week), teachers should decrease the need for themselves to model and provide strategies as they work to increase students’ understanding, ease, speed, and accuracy in decomposing and composing numbers.

Assessment

To assess students through number talks, teachers need to collect data while delivering the number talk. Through listening to students’ explanations and numerical strategies for arriving at solutions, teachers can conduct formative assessments. If a student is unable to articulate a strategy and/or is incorrect in their solution, teachers can use these data to inform interventions and supports for mathematical skills and knowledge for that student, including prior knowledge thought obtained, as well as deciding what would be an appropriate next number talk. For example, in presenting upper elementary students with high-incidence disabilities a number talk based on place value, a teacher can determine whether students struggle with this foundational concept and, if so, address the concept in their special education class or support time. With the number talk, “Which number is greater: 57 or 4 tens and 16 ones?” a teacher can assess whether students have sufficient understanding of place value, which is needed for them to make sense of addition and subtraction with regrouping. Similarly, in presenting a fraction magnitude number talk to middle school students with high-incidence disabilities (e.g., “Is 3/8 greater than or less than $1 / 2$ , and explain how you know”), a special education teacher can conduct a formative assessment of a student’s fundamental knowledge of fractions, which is needed when engaging with the concepts of operations (+, −, ×, and ÷) with fractions.

One way in which teachers can collect data through number talks is by noting which correct or common incorrect strategies or solutions students shared or presented (see Tables 1 and 2). With the smaller size of a special education setting for number talk delivery, a teacher can keep a hard copy—or electronic—data collection sheet handy and indicate whether students achieve the correct answer, the number of strategies they indicate they thought of per holding up their number of fingers for number of strategies, as well as the identifying what strategies students shared—both common correct and incorrect ones. By using number talks to formatively assess students and keeping data from the formative assessments, special education can apply number talk data for Individualized Education Program (IEP) goals and to determine interventions.

Table 1.

Sample Formative Assessment Data Collection for Number Talks: Fourth Grade.

Fourth-Grade Resource Room Math ClassNumber Talk: Which Number Is Greater: 57 or 4 Tens and 16 Ones?
Student	Determined Which Is Greater: 57 or 4 Tens and 16 Ones	No. of Strategies Shown	Incorrect Answer (57 > 16)	Incorrect Answer (57 > 20)	Incorrect Answer (57 < 416)
Jacob	N	1	Y	N
Jillian	N	1	N	Y
Kate	Y	1	—	—
Kaden	N	0	N	N
Raymond	N	1	N	N

Table 2.

Sample Formative Assessment Data Collection for Number Talk: Third Grade.

Third-Grade Resource Room Math ClassNumber Talk: Olivia Had 19 Football Cards. She Just Got a New Pack of Football Cards With 18 Cards in the Pack. How Many Football Cards Does Olivia Have Now?
Student	Determined 37 Cards	No. of Strategies Shown	Strategy: “Just Knew” (Did Mental Math)	Strategy: Counting On (i.e., Counting Up by 18 From 19	Strategy: Breaking Numbers Up and Counting on by Chunks (i.e., 10 + 10 + 17 [8 + 9])	Strategy: Rounded Both to 20 and Subtracted (i.e., 20 + 20 − 1 − 2)
Abby	Y	1	N	Y	N	N
Charlie	Y	2	Y	N	Y	N
Andrew	N	1	N	Y	N	N
Luca	N	0	—	—	—	—
Preston	N	1	N	N	N	Y

Scaffolds for the general education setting

To prepare students with high-incidence disabilities to actively engage and equitably participate in general education number talks as well as activate students’ prior knowledge for general education settings, teachers need to provide students many opportunities with this practice in special education settings. When using number talks and presenting or rephrasing numerical strategies, special education teachers need to use consistent and appropriate (i.e., precise) mathematical language (e.g., using “greater” than rather than “bigger,” “numerator” as opposed to “top number” of a fraction, and “regrouping” in place of “borrowing” when subtracting; Hughes et al., 2016; Karp et al., 2014). The consistent use of appropriate language could be supported through anchor charts in the classroom. Similarly, the focus on rules, and particularly rules that expire (e.g., bigger numbers cannot be subtracted from smaller numbers and multiplying by 10 adds a zero to the end the number), should not be supported or taught during number talks (Karp et al., 2014). Not only do these rules hurt students in the long run, but they also distract from students’ understanding of place value and number sense. Finally, teachers can use number talks to help develop a growth mindset in students with high-incidence disabilities; that is, believing that by trying out ideas, struggling, and working with mental mathematics, they can improve their understanding (Dweck, 2008, 2015). One means of helping students develop a growth mindset in mathematics is through self-regulation, such as helping students to set goal and then work to achieve their goal (Wang et al., 2019).

To select number talks to support students with high-incidence disabilities in general education settings, teachers need to know the scope and sequence of the mathematical content in the general education setting. By knowing what general education teachers are addressing in mathematics classes, special education teachers can purposively select number talks that connect to precursor skills or background knowledge. As stated, special education teachers should implement numbers talks in a consistent, although appropriate to the spirit of number talks, manner as their students’ general education teachers, albeit with the aforementioned accommodations and scaffolds needed at the beginning. By preparing students for the content and the process, students with disabilities may feel more at ease with number talks in a larger setting and ultimately more willing to participate in the general education setting.

Conclusion

Although there is limited research regarding number talks for students with high-incidence disabilities and their use within special education mathematics settings by special education teachers, teachers supporting students with high-incidence disabilities are encouraged to explore the use of number talks. Why? Because, number talks are increasingly occurring in elementary and secondary general education settings. By engaging elementary and secondary students with high-incidence disabilities in number talks in small group settings, special education teachers may be able to prepare their students to more actively engage in number talks in the larger general education setting. Students with disabilities may be able to become familiar with the process of mentally composing and decomposing numbers as well as orally sharing their mathematical thinking.

Special education teachers are also encouraged to consider using number talks as both intervention and assessment. In terms of assessment, number talks may provide special education teachers a quick means of engaging in formative assessment of their students’ mathematical thinking, understanding of number and number relationships, mental computation, and strategies to solve problems. Special education teachers can then use their assessment of students from number talks to design interventions, including future number talks. In terms of intervention, number talks may develop and expand students’ fluency and understanding of number, number relationships, and operations and computation. They may also work to both activate students’ background or prior knowledge to enable them to participate and understand concepts presented in their general education mathematics classes. By routinely and consistently implementing number talks, students with high-incidence disabilities may improve in their knowledge and skills as well as their flexibility and accuracy with numbers and operations.

While number talks should be implemented with fidelity, special education teachers should consider implementing them with flexibility. As discussed in this article, special education teachers are encouraged to expand the norms of number talks to allow students to use manipulatives, pictorial representations, or paper-and-pencil with talks, and then work to systematically fade these supports. Finally, special education teachers are also urged to clarify misconceptions or inaccuracies students present during number talks and explicitly teach students with high-incidence disabilities efficient and effective numerical strategies when and where needed.

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.

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