Effective and Efficient Mathematics Instruction for Students Receiving Special Education Services

Like developing proficient reading skills, developing competence in mathematics is essential to an individual’s success in school and post-secondary endeavors. At school, a student’s success in higher-level mathematics (e.g., algebra, geometry, calculus) depends on her mastery of basic computation and problem-solving skills (Powell, 2019). Outside of school, mathematics competence impacts one’s ability to engage in recreation and leisure activities (e.g., play board games, pick up the tab at a restaurant), live independently (e.g., pay rent and utilities, maintain a bank account,  manage a budget, shop for groceries), and obtain and maintain employment (e.g., make change as a server, inventory items in a warehouse) (Jimenez & Saunders, 2019).

A starting point for the presentation of effective mathematics instruction to students who demonstrate persistent and significant academic achievement deficits in this content area is a consideration of six matters that Powell (n.d.) identified as central to these students’ mathematics programming.

1. Incorporate systematic and explicit instruction.
2. Focus on the vocabulary used to explain mathematics, as well as the symbols that are used in mathematics.
3. Use the concrete-representational-abstract framework (CRA Framework) to give the students opportunities to use hands-on materials, pictorial representations, and representations representing the abstract. Abstract representations consist of the numerals and symbols of mathematics.
4. Use fluency-building activities, meaning those activities that build fluency with (a) mathematics basic facts and (b) computational algorithms.
5. Engage in effective questioning strategies.
6. Conduct an error analysis of students’ work and use this information to inform instruction.

Dr. Sarah Powell, a professor at the University of Texas, presents the online modules comprising the Outside Activities featured in this chapter. She is an expert in the area of mathematics intensive instruction. Thus, I feel very fortunate that we can hear directly from her. This arrangement is like having a nationally renowned guest speaker present to our class.

Dr. Powell is very informative during her presentations, and her content can be applied immediately to any Tier 3 mathematics instruction you present. Moreover, the content she presents can be applied immediately to any mathematics instruction any teacher presents.

According to The IRIS Center (2017), high-quality, Tier 1 mathematics instruction consists of two elements: (a) a standards-based curriculum consisting of relevant content, such as the content and process standards identified by the National Council of Teachers of Mathematics (NCTM) and (b) a teacher’s use of evidence-based practices. Accordingly, after explaining why mathematics intensive intervention is important, Dr. Powell addresses numerous topics pertaining to what to teach (i.e., curriculum content) and how to teach it (i.e., which instructional strategies to use). She alludes to the fact that intensive intervention is intended for the 3%-5% of students who are the lowest performing students in a school concerning the subject matter area in which they are experiencing difficulty. They demonstrate an academic achievement deficit even after receiving high-quality instruction as a part of the Tier 1 and Tier 2 services they were provided.

Dr. Powell makes a very important point when she says that, during intensive intervention, you cannot reteach all of the content a student should have mastered earlier. Thus, educators must make a concerted effort to identify which mathematics content needs to be taught. For instance, she highlights how teaching Base 4 content is a non-example of what should be taught. Conversely, she identifies the following as mathematics areas where students must have mastery:

• Fluency
• Place value
• Operations
• Problem-solving

The information presented by Dr. Powell highlights the primary distinction that needs to be made concerning special education being accommodations or special education being intensive intervention. Special education as accommodations would proceed with teaching the grade-level mathematics content for the standards highlighted in the core curriculum’s scope and sequence. It would primarily rely on accommodations to enable the student to master the content. Special education as intensive intervention acknowledges that a student with a disability who needs to receive this intervention is performing below the grade-level standard, which means the student needs to work to master content that should have been mastered previously. This circumstance highlights several issues, such as some asserting that it takes a 3rd or 4th-grade student four months of instruction to master a targeted learning outcome that could have been mastered in 20 minutes of effective instruction when the student was in kindergarten.

References

Jimenez, B., & Saunders, A. (2019). Increasing efficiency in mathematics: Teaching subitizing to students with moderate intellectual disabilities. Journal of Developmental and Physical Disabilities, 31(1), 23-37. doi: 10.1007/s10882-018-9624-y

National Center on Intensive Intervention (n.d.). Part 2: What mathematical content do students need to master across kindergarten through eighth grade? Retrieved from https://www.youtube.com/watch?v=r1TfAzqcg6M

Powell, S. (2019). Part 1: Why is mathematics intensive intervention important? (Retrieved November 10, 2022 from https://www.youtube.com/watch?v=rYy6yb_rWM0&t=1s

Powell, S. (n.d.). What should we consider when teaching students with math difficulties? Retrieved April 18, 2023 from https://intensiveintervention.org/resource/what-should-we-consider-when-teaching-students-math-difficulty

The IRIS Center. (2017). High-quality mathematics instruction: What teachers should know. Retrieved from https://iris.peabody.vanderbilt.edu/module/math/

Chapter Primer

The information presented below serves as an introduction to the content presented in this chapter. This content comes from a series of presentations by Dr. Powell. The titles of these presentations and the content addressed in them are as follows:

• “Why is mathematics intensive intervention important?” Dr. Powell discusses why mathematics intensive intervention is warranted and warranted as soon as a student demonstrates difficulty mastering grade-level content.
• “What mathematical content do students need to master across kindergarten through eighth grade?” Dr. Powell discusses the mathematics content that should be the focus of intensive instruction, emphasizing that it is impossible to reteach all the content a student who needs intensive instruction has not mastered.
• “How do you use explicit instruction within intensive intervention?” Dr. Powell discusses employing an explicit instruction approach to present mathematics intensive instruction.
• “How should multiple representations be used?” Dr. Powell discusses how to use concrete, representational, and abstract depictions of mathematical concepts to enhance the effectiveness of intensive instruction.
• “How do you attend to language within intensive intervention?” Dr. Powell presents numerous examples of the precise mathematics language one should use when presenting intensive instruction. She also emphasizes the use of concise language.
• “How do you build fact fluency within intensive intervention?” Dr. Powell compares how fluency with math facts serves the same function as fluency with decoding words, which is to free up the student’s cognitive capacity to be able to focus on higher-level tasks.
• “How do you incorporate effective problem solving within intensive intervention?” The content in this video addresses a question many UWF students pose: How do you use an explicit instruction approach to teach mathematics beyond basic skills acquisition?
• “How do you incorporate a motivational component in intensive intervention?” While research has revealed that a student’s (a) success and (b) understanding of the value of learning mathematics content lead to motivation (Archer, 2021), in her video, Dr. Powell discusses various strategies that involve the use of external contingencies.

References

A. Archer (2021, February 8). The importance of explicit reading instruction [EDVIEW360 Podcast]. Retrieved May 4, 2023  from https://voyagersopris.buzzsprout.com/

Why is mathematics intensive intervention important?

Watch this module from the National Center on Intensive Intervention titled “Why is mathematics intensive intervention important?” to gain an understanding of the reasons why mathematics intensive intervention is not only warranted but warranted as soon as a student demonstrates difficulty mastering grade-level mathematics content.

You can stop watching the module at the 18:11 mark (i.e., 18 minutes, 11 seconds).

Note that you do not have to do any workbook activity, journal activity, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about mathematics intensive intervention.

After working through this module, you should be able to

• Discuss the relationship between low math scores in early grades (i.e., Preschool and Kindergarten) and later grades (i.e., High School)
• Explain what is meant by saying that “math is predictive”  when examined both at elementary and middle school grades
• State whether the following is True or False: Some predictive studies have shown that how a student performs at math while in school predicts outcomes after high school.
• State whether the following is True or False: Research demonstrates the need for timely, effective intervention.
• State whether the following is True or False: There are numerous studies that show when you provide intensive intervention in mathematics, it increases the mathematics scores and the mathematical pathways of students.
• State whether the following is True or False: Both school and adult outcomes are related to your math performance during school.

What mathematical content do students need to master across kindergarten through eighth grade?

Watch this module from the National Center on Intensive Intervention titled “What mathematical content do students need to master across kindergarten through eighth grade?” to gain insight into the mathematics content that should be the focus of intensive intervention.

You can stop watching the module at the 19:37 mark (i.e., 19 minutes, 37 seconds).

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about the main topic addressed in this module.

After working through this module, you should be able to

• Discuss, in general, what math content is the focus of intensive intervention
• Identify the areas that Dr. Powell believes are really important for students to have mastery (i.e., comprise what she believes are the foundational skills of mathematics)
• State whether the following is True or False: Our instruction [meaning intensive instruction] should always be related to the scope and sequence of mathematics that the student needs to learn.

How do you use explicit instruction within intensive intervention?

Watch this module from the National Center on Intensive Intervention titled “How do you use explicit instruction within intensive intervention?” to learn how explicit instruction is used within intensive intervention that is designed to teach mathematics to students who exhibit a persistent and significant academic achievement deficit in this subject matter area.

You must watch the entire module (Length: 41 minutes, 17 seconds). 

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about the main topic addressed in this video.

After working through this module, you should be able to

• Complete the following sentence: Regarding the statement of a clear objective, Dr. Powell says you should state the goal of the lesson and why ____________________.
• Complete the following sentence: When you model the steps of the problem you should be precise and concise, meaning use ____________________ language.
• List examples of the type of language referred to in the previous bulleted item
• Explain the two types of planned examples, to include the reason for using both types
• List the four items Dr. Powell identified as Supporting Practices
• Discuss whether Supporting Practices are used during Modeling, Practice, or both of these elements/components of the Explicit Instruction Framework
• Discuss the two types of questions, lower-level and higher-level, that teachers should ask
• List a minimum of three ways students can respond during intensive intervention in math
• Discuss the purpose of requiring students to respond frequently
• Differentiate between affirmative and corrective feedback
• Describe the two types of affirmative feedback
• Explain the importance of providing corrective feedback
• Explain how Dr. Powell defines the concept of “a brisk pace”
• Discuss why Modeling and Practice might not take place, equally, during a math lesson that is based on explicit instruction

How should multiple representations be used?

Watch this module from the National Center on Intensive Intervention titled “How should multiple representations be used?” to obtain an understanding of using what are referred to as multiple representations when presenting mathematics intensive intervention. Two of these representations, concrete and representational, enable students to develop an understanding of the concepts that are addressed with the use of the abstract symbols involved in mathematics. 

You must watch the entire module (Length: 29 minutes, 05 seconds).

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about the main topic addressed in this video.

After working through this module, you should be able to:

• List the three components of the CRA framework (CRA=Concrete, Representational, Abstract)
• Discuss why a teacher uses multiple representations to teach mathematics
• Explain what is meant by the following: “The CRA framework should be thought of as overlapping rather than sequential.”
• Describe “the concrete of mathematics,” particularly what it is and what are some examples
• Describe “the representational of mathematics,” particularly what it is and what are some examples
• Explain how the representational in mathematics is related to the concrete
• Differentiate between prepared representationals and other representationals
• Describe “the abstract of mathematics,” particularly what it is and what are some examples
• Note whether words are an abstract form of mathematics
• Identify which form – concrete, representational, abstract – students use most often to solve mathematics

How do you attend to language within intensive intervention?

Watch this module from the National Center on Intensive Intervention titled “How do you attend to language within intensive intervention?” to learn about the type of language a teacher should use when presenting mathematics intensive intervention.

You must watch the entire module (Length: 27 minutes, 56 seconds).

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge using proper language when presenting mathematics intensive intervention.

After working through this module, you should be able to

• Explain how a teacher’s precise use of math language is related to the math vocabulary presented on a high-stakes assessment
• Discuss the meanings of the terms “homonyms” and “homophones” with respect to math language
• Define the four types of terms students have to know in math language: technical terms, subtechnical terms, symbolic terms, and general terms
• Identify examples of each of the four types of terms students have to know in math language
• Explain what is meant when a teacher is told to “present precise and concise” math language

How do you build fact fluency within intensive intervention?

Watch this module from the National Center on Intensive Intervention titled “How do you build fact fluency within intensive intervention?” to learn information you can use in mathematics intensive intervention to address the development of a student’s basic fact fluency.

You must watch the entire module which is 55:09 in length (i.e., 55 minutes, 09 seconds). 

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about the main topic covered in this video.

After working through this module, you should be able to

• State why it is important to work on addition, subtraction, multiplication, and division basic facts
• State the definition of an addition basic fact
• State whether the following is True or False: In the elementary grades we should refrain from using complex math language that includes words such as “addends” and “sum.”
• State the definition of a subtraction basic fact
• Which term – minuend, subtrahend, or difference – is regularly expected for students to know
• State three things we always want to help students understand related to addition and subtraction facts
• Explain the two concepts that pertain to addition
• Explain the two concepts that pertain to subtraction
• Describe the Counting Up strategy
• Discuss which addition facts students should learn how to solve using the Counting Up strategy
• State whether the following is True or False: According to Dr. Powell, students who exhibit difficulties learning math have an equally difficult time counting forward and counting backward.
• Discuss how to use a Counting Up strategy to solve a subtraction problem
• State the meaning of the term “minus number”
• State whether the following is True or False: Fluency applies to addition and subtraction basic facts.
• State whether the following is True or False: Fluency applies to multiplication and division basic facts.
• State the definition for a multiplication basic fact
• State whether the following is True or False: Students should know both the term “factor” and “product.”
• Define the terms “factor” and “product”
• State the definition for a division basic fact
• Define the terms “dividend,” “divisor,” and “quotient”
• State the three areas where students need to develop knowledge related to multiplication and division facts
• Discuss the two ways of thinking about multiplication
• Describe the two ways of thinking about division
• State whether the following is True or False: If students understand their multiplication facts, a teacher can use the inverse relationship with multiplication and division to enable students to understand their division facts.
• Identify an example of the commutative property of multiplication
• Identify an example of the inverse relationship between multiplication and division
• Discuss the use of a multiplication table for the purpose of teaching students how to solve multiplication and division facts
• State whether the following is True or False: The fluency strategies described by Dr. Powell are only applicable to addition and subtraction basic facts.
• Fluency activities are to be brief and ____________________ to implement.
• State whether the following is True or False: Fluency activities are to be implemented daily within your intensive intervention.
• Describe each of the following fluency building activities: (a) Copy, Cover, Compare; (b) Taped Problems; (c) Flash Cards; (d) Magic Squares; (e) Roll Dice/Dominoes/Playing Cards
• Describe the Incremental Rehearsal strategy with flashcards
• Discuss considerations concerning using technology to practice fact fluency
• Explain the relevance of each of the following to teaching mathematics basic facts: (a) Fact practice should occur regularly, (b) Fact practice should be brief, (c) Practice the facts students need to learn, and (d) Do not practice all the facts at one time
• List the three primary topics that were addressed in this video with respect to the question, “How do you build fact fluency within intensive intervention?”

How do you incorporate effective problem-solving strategies in intensive intervention?

Watch this module from the National Center on Intensive Intervention titled “How do you incorporate effective problem-solving strategies in intensive intervention?” to gain insight about how to teach problem-solving in mathematics intensive intervention.

You must watch the entire module which is 1:02:54 in length (i.e., 1 hour, 02 minutes, 54 seconds). 

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about the main topic covered in this module.

After working through this module, you should be able to

• State two reasons for focusing on math problem-solving
• State whether the following is True or False: Word problems are the only types of math problems students have to solve.
• State the reasons why students with mathematics difficulty find word problem-solving very challenging
• Explain why Dr. Powell said, “We know one thing about teaching problem-solving ineffectively, and that is to use key words.”
• Explain what is meant by the saying that “teachers are always trying to help students understand the mathematics continuum”
• Describe an attack strategy
• Describe a routine word problem and an instructional word problem
• List eight attack strategies
• State the idea with any attack strategy
• Identify the primary thing that students do NOT do with respect to a word problem
• State whether the following is True or False: You must use the same attack strategy with all of your students receiving intensive instruction.
• Define the term “schema” with respect to word problems
• State how a teacher helps students develop a deep understanding of problem solving
• Differentiate between the Additive Schemas and Multiplicative Schemas
• Complete this sentence: Dr. Powell likes to describe word problems as a mix of ___________________ and ___________________.
• Explain one way to use the UPSCheck strategy
• Differentiate between the basic elements/components of a Total Problem, Difference Problem, and Change Problem
• Discuss teaching problem solving by the deep structure of the problem rather than problem-solving by operations
• Differentiate between the basic elements/components of an Equal Groups problem, a Comparison problem, and a Ratios/Proportions problem
• Discuss each of these considerations with respect to word problems: (a) Many word problems actually combine schemas, (b) Sometimes students don’t answer word problems with an answer, and (c) Not all word problems are routine word problems (i.e., they are instructional word problems)
• State whether the following is True or False: Problem-solving should be taught with explicit instruction.
• State whether the following is True or False: Problem-solving instruction should be provided regularly.

How do you incorporate a motivational component in intensive intervention?

Watch this module from the National Center on Intensive Intervention titled “How do you incorporate a motivational component in intensive intervention?” to learn about aspects of a motivational component to incorporate into mathematics intensive intervention.

You must watch the entire module (Length: 7 minutes, 20 seconds).

Note that you do not have to do any workbook activity, journal activity, outside reading, etc. that is referred to in the module. However, you should listen to the discussions about these activities since this information will extend your knowledge about incorporating a motivational component in mathematics intensive intervention.

After working through this module, you should be able to

• Identify these three evidence-based strategies you can use regarding the delivery of an instructional platform: (a) explicit instruction, (b) multiple representations, and (c) concise language
• Identify these three strategies that should be embedded within the instructional platform: (a) fluency building, (b) problem solving instruction, and (c) a motivation component
• Discuss why there is a need for a motivation component
• Describe an example of a motivation component that is a match with what is going on in the intervention
• Explain what is meant by this statement: A motivational component is really meant to help students regulate their own behavior.
• State whether the following is True or False: A motivational component is necessary for all students.

No Cost Resources Pertaining to This Topic

Each item below is a no cost resource that presents information about some aspect of this chapter’s focus, which is presenting effective and efficient mathematics instruction to students receiving special education services. To access a resource, use the link provided.

High-Quality Mathematics Instruction: What Teachers Should Know This online module, which is available from The IRIS Center, describes what the authors have identified as the two essential components of high-quality mathematics instruction: a standards-based curriculum and evidence-based practices. More specifically, the module highlights a number of evidence-based practices as well as other classroom practices that teachers can use to teach mathematics.

What Should We Consider When Teaching Students With Math Difficulties? In this video from the National Center on Intensive Intervention, Dr. Sarah Powell discusses six things to consider when teaching students with math difficulties:  (a) incorporate systematic and explicit instruction, (b) focus on vocabulary, (c) use the concrete-representational-abstract framework, (d) use fluency building activities, (e) engage in effective questioning strategies, and (f) conduct an error analysis of students’ work. This video is also available on YouTube at  (https://www.youtube.com/watch?v=dQX9Cl0s04I&t=1s) and is 8 minutes, 11 seconds long.

Why is mathematics intensive intervention important? This module from the the National Center on Intensive Intervention explains reasons why mathematics intensive intervention is not only warranted, but warranted as soon as a student demonstrates difficulty mastering grade-level mathematics content.

What mathematical content do students need to master across kindergarten through eighth grade? This online module is from the National Center on Intensive Intervention’s webpage titled “Developing a Scope and Sequence for Intensive Math Intervention.” On this webpage, this online module is Part 2. 

How do you use explicit instruction within intensive intervention? This module from the National Center on Intensive Intervention reviews the use of explicit instruction to teach mathematics. Teachers learn modeling begins with a statement of the goal and importance of learning a skill in mathematics. Modeling is explained as a step-by-step overview of mathematics skills with meaningful examples and non-examples. Teachers also learn about the use of guided and independent practice. Emphasis is placed on the need for high- and low-level questions, frequent responses, adequate feedback, and maintaining a brisk pace during both modeling and practice.

How do you build fact fluency within intensive intervention? This module from the National Center on Intensive Intervention presents information about the use of mathematics intensive intervention to address the development of students’ basic fact fluency.

How should multiple representations be used within intensive intervention?” This module from the National Center on Intensive Intervention highlights the use of multiple representations to enhance the delivery of the instructional platform. Teachers learn the importance of using concrete tools (i.e., manipulatives) and pictorial representations to help students understand the numbers and symbols (i.e., abstract) of mathematics.

How do you attend to language within intensive intervention? In this module, from the National Center on Intensive Intervention, Dr. Sarah Powell describes the importance of using formal mathematics language in intensive intervention. Teachers (a) review precision with mathematics language and (b) work through examples in which formal language could replace informal language.

How do you incorporate effective problem-solving strategies in intensive intervention? This module from the National Center on Intensive Intervention discusses how to teach problem-solving in mathematics intensive intervention.

How do you incorporate a motivational component in intensive intervention? This module from the National Center on Intensive Intervention discusses aspects of a motivational component to incorporate into mathematics intensive intervention.

Making Fractions Make Sense: Considerations for Secondary and Intensive Intervention  This webinar from the National Center on Intensive Intervention is presented by Drs. Russell Gersten, Sarah Powell, and Robin Finelli Schumacher, who discuss (a) the importance of fractions instruction and typical challenges faced by students, (b) recommendations for fractions instruction, and (c) considerations for supporting students within secondary or Tier 2 intensive intervention.

Progress Monitoring: Mathematics This online module from The IRIS Center introduces users to progress monitoring in mathematics, which is a type of formative assessment in which student learning is evaluated to provide useful feedback about performance to both learners and teachers (Estimated completion time: 2 hours).