Unlike students who struggle with reading, students who struggle with math often do not have the benefit of response to intervention (RtI) in their elementary and middle schools.
A recent What Works Clearinghouse Practice Guide (Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools) aims to change that by providing educators with 8 recommendations for implementing RtI in math.
The recommendations were developed by an expert panel chaired by Russell Gersten of the Instructional Research Group in Calif. Gersten is a research mathematician active in issues related to K-8 math education. The panel also included 2 professors of math education, special educators, and a math coach. The panel worked collaboratively to develop recommendations based on the best available research evidence and its own expertise in math, special education, research and practice.
“To date, little research has been conducted to identify the most effective ways to initiate and implement RtI frameworks for mathematics,” the panel writes. “However, there is a rich body of research on effective mathematics interventions implemented outside an RtI framework.”
Below are the panel’s 8 recommendations for implementing RtI with low-achieving students:
- Screen all students
- Focus on whole numbers for interventions in grades K-5 and on rational numbers for grades 4-8.
- Provide explicit and systematic instruction to struggling students
- Teach students to look for the common underlying structures of word problems.
- Use visual representations of mathematical ideas.
- Devote about 10 minutes in each intervention session to building fluent retrieval of basic arithmetic facts.
- Monitor the progress of students receiving supplemental instruction and other students who are at risk.
- Include motivational strategies in tier 2 and tier 3 interventions.
1. Screen all students.
Why screen students who are doing fine? To develop a distribution of achievement from high to low, the panel says. If students who are not considered at risk are not screened, the distribution would consist only of at-risk students. “This could create a situation where some students at the ‘top’ of the distribution are in reality at risk but not identified as such,” the panel says. Another reason is to make sure those students stay on track, the panel says.
No one screening measure is perfect. Screening measures target what is referred to as “number sense.” Some measures focus on only one aspect of number sense (such as magnitude comparison) and others assess 4-8 aspects of number sense. Single-component measures take as little as 5 minutes to administer while multi-component measures take as much as 20 minutes.
Single-component tests with the best ability to predict students’ mathematics performance focus on magnitude comparison and/or strategic counting. The broader measures seem to predict number sense with slightly greater accuracy, but consider efficiency so that you can screen many students in a short time. Use the same screening tool across the district to analyze results across schools.
Set up a team to select a screening measure. Most research focuses on valid screening measures for K-2 students, but there are also reasonable research-based strategies for students in higher grades, according to the panel. Screening should occur in the beginning and middle of year. In grades 4-8, use screening data in combination with state testing results. The previous year’s state test results can be used as the first screen for students, for example.
2. Focus on whole numbers for grades K-5 and on rational numbers for grades 4-8.
Cover fewer topics in more depth with students, especially those who struggle with mathematics. Instead of brief ventures into many topics in the course of a school year, emphasize instruction in whole numbers and rational numbers. Both the National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points (2006) and the National Mathematics Advisory Panel (NMAP) have endorsed this approach for all students, but it is especially important for students struggling with math.
Appoint committees to review instructional materials for interventions. Include mathematicians with a knowledge of elementary and middle school math curriculum and experts in mathematics instruction, The committees should assess how well intervention materials meet 4 criteria. They should ask if the materials:
- combine computation with problem- solving and pictorial representations (rather than teach problem-solving and computing separately)?
- emphasize the reasoning underlying calculation methods and focus the students’ attention on making sense of the mathe matics?
- ensure that students build algorithmic proficiency?
- include frequent review for both consolidating and understanding the links of the mathematical principles?
The intervention program should include an assessment for appropriate placement in interventions. Building students’ foundational proficiencies with the interventions is more critical than alignment with the core curriculum, the panel says. Educators should not feel that they need to cover every topic in the intervention materials. Students will be exposed to supplemental topics such as measurement and data analysis in the general classroom.
3. Provide explicit and systematic instruction to struggling students.
The National Mathematics Advisory Panel defines explicit instruction as follows:
- Teachers provide clear models for solving a problem type using a variety of examples.
- Students receive extensive practice in use of newly learned strategies and skills.
- Students are provided with opportunities to think aloud (they talk through the decisions they make and the steps they take).
- Students receive extensive feedback.
Struggling students should receive explicit instruction regularly. That instruction should ensure that students will have the foundational skills and conceptual knowledge they need for grade-level mathematics. Scaffolded practice, in which the teacher transfers control of problem-solving to the student, was a component in 4 of the 6 studies reviewed by the panel on this recommendation.
Instructional materials should systematically build proficiency by gradually and logically introducing concepts. They should provide sample think-alouds and scenarios for explaining concepts and working through operations. A major flaw in many instructional materials is that they provide only one or two models of how to approach a problem and that most of these models are for easy-to-solve problems, the panel writes. If the instructional materials do not have enough samples and scenarios, consider using a math coach or specialist to develop a template of an effective lesson.
Educators should observe an intervention before conducting one by watching a video as part or getting hands-on experience with colleauges or students. Trainers can observe these practice runs and provide feedback. Professional development also should provide educators with in-depth knowledge of the mathematics content that is to be covered in the intervention.
4. Teach students to look for the common underlying structures of word problems.
Students who have difficulty with math typically need a lot of help with word problems. Focus on helping students understand the underlying structure of word problems. Much like stories, problems give meaning to mathematical operations such as addition and subtraction.
“When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems,” the panel writes.
Teach students about the structure of various problem types so they can categorize them and determine appropriate solutions for that type of problem. They need to be able to distinguish between addition and subtraction problems and other types of problems.
Students can sometimes miss the common structure between old and new problems because of superficial differences in details. “Teachers should explain these irrelevant superficial features explicitly and systematically,” the panel writes. Have students practice sets of problems with varied superficial features.
Curricular material may not classify problems into problem types. A math coach or specialist can develop an instructional sequence for teaching problem types to students. As problems get more complex, the task of discriminating among problem types becomes more difficult, the panel says. Interventionists will need high-quality professional development to help them teach students how to distinguish among types of problems that often involve multiple steps.
5. Use visual representations of mathematical ideas.
Pie charts, number lines, graphs and simple drawings of concrete objects such as blocks or cups or simplified drawings help students understand the relationships between visual representations and the abstract symbols of mathematics. Expressing mathematical ideas with visual representations and converting visual representations into symbols is critical for success in math, the panel writes. Typically, exposure to visual representations is unsystematic and insufficient. “We recommend that interventionists use such abstract visual representations extensively and consistently,” the panel writes. Visual representations help scaffold an understanding of the abstract symbols of math, the panel writes.
Use manipulatives as a stepping stone only, the panel says. The goal is to help students reach an understanding of the abstract level as soon as possible.
Pictures of objects are commonly used to teach addition and subtraction in the lower grades, but in upper grades diagrams and pictures also can be used to teach fractions and the basic structure of word problems. The goal of using a number line should be for students to create a mental number line and to know how to move along the line according to the marking arrows.
Use of concrete objects such as blocks in early elementary grades should be used even more extensively to reinforce students’ understanding of basic concepts and operations. They should be used in the upper grades to help students understand math at a more abstract level.
“Using consistent language across representational systems (manipulatives, visual representations, and abstract symbols) has been an important component in several research studies,” the panel writes.
6. Devote about 10 minutes in each intervention session to building fluent retrieval of basic arithmetic facts.
Teachers and texts often assume students’ automatic retrieval of facts such as 3 x 9= ___ and 11 — 7=___. Inability to automatically retrieve this information impedes understanding of the concepts students encounter. Beginning in grade 2, the panel recommends spending about 10 minutes during each intervention session to help students practice automatic retrieval of basic arithmetic facts.
An efficient way to work on automatic retrieval is to integrate previously learned facts into practice activities. To reduce students’ frustration with learning new material, educators can individualize practice sets so students learn 1 or 2 new facts, practice several recently acquired facts and review previously learned facts.
Students in K-2 may need explicit instruction in the “counting-up strategy”, a strategy many students develop on their own, sometimes as early as age 4. But students with difficulties in math tend not to develop this strategy on their own. To solve 3 + 5=__, for example, students can be explicitly taught to find the smaller number, hold up 3 fingers then count up an additional 5 fingers until they amount to 8.
In grades 2-8, students should learn to use number properties (associative, distributive and commutative) instead of just relying on rote memorization. For example, to compute the seemingly difficult multiplication of 13 X 7=__, students should be reminded that 13 =10 + 3 and that the computation is simpler if broken down to 13 X 7 =(10 + 3) X 7=10 X 7+3 X 7.
7. Monitor the progress of students receiving supplemental instruction and other students who are at risk.
At least once a month, monitor the progress of tier-2 and tier-3 students and borderline tier-1 students who perform just above the cutoff on the general outcome measures. Students should be regrouped if they subsequently meet benchmark or fall behind. Beyond regrouping, progress monitoring data should be used to determine when instructional changes are needed.
Assessment results can be used to determine which concepts need to be reviewed, which need to be re-taught, and which have been mastered. Curriculum-embedded assessments are often administered daily for students in kindergarten and grade 1 and biweekly for students in grades 2 through 6. These assessments usually do not have the same high technical characteristics as the general outcome measures, so the panel recommends using both types of measures. Interventionists need to be cautious about assuming that mastery of individual skills and concepts will translate into improvements in overall proficiency.
If teachers are too busy to assess student progress with monitoring measures, consider training paraprofessionals or other school staff to do so. Also, students from different classrooms can be placed in the same tier-2 and tier-3 intervention groups if students within one class are at very different levels.
8. Include motivational strategies in tier- 2 and tier-3 interventions.
Do not forget how challenging basic arithmetic is for struggling students. Include behavioral supports that promote student effort. Offer rewards for engagement, persistence (completion of tasks) and achievement and performance. Without such supports, even a well-designed intervention may falter. Praise students for the effort and engagement. Praise should be immediate and specific. Stay away from generic and empty praise (“good job” or “keep up the good work”).Praise is most effective when it points to specific progress that a student has made and recognizes students’ actual effort.
Give rewards for completion of math tasks and accurate work. Educators can contact students’ parents to inform them of their children’s successes by phone or email or in a note. Take into account each student’s interests before choosing an appropriate reward. Allow students to chart their progress and set goals for improvement. This can help them develop self-regulated learning by encouraging them to take greater responsibility for themselves.
“Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools, Institute of Education Sciences Practice Guide, National Center for Education Evaluation and Regional Assistance, April 2009.