RTI and Math Instruction
by Amanda VanDerHeyden, Ph.D., Education Research and Consulting, Inc., Fairhope, AL
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Using RTI to Improve Learning in Mathematics
Response to Intervention (RTI) has become a vehicle for system reform because it provides a framework in which data can be relied on as the basis for making relative judgments (e.g., determining who needs help the most and how much they need) and for distributing instructional resources to promote the greatest good for the greatest number of students. Much of the writing and research on RTI has occurred in the area of reading, but RTI is not limited to reading. Rather, it is a science of decision making that can be applied to a variety of “problem” behaviors. RTI, properly understood and used, is focused on improving student learning. Ensuring the development of mathematics competence during the primary grades is essential to later learning success. Key findings in the literature highlight the need to focus on early mathematics instruction:
 Children who have had less experience or exposure to mathematical concepts and numeracy are at high risk for mathematics failure (Griffin & Case, 1997).
 Most students fail to meet minimal mathematics proficiency standards by the end of their formal schooling (U.S. Department of Education, 2003).
 Students identified with specific learning disabilities perform lower and grow at a slower pace relative to their peers in learning mathematics.
 Existing instructional tools and textbooks often do a poor job of adhering to important instructional principles for learning in mathematics (National Mathematics Advisory Panel, 2008).
 Math is highly proceduralized and continually builds on previous knowledge for successful learning. Hence, early deficits have enduring and devastating effects on later learning, as indicated in The Head Start Path to Positive Child Outcomes (U.S. Department of Health and Human Services, 2001) and elsewhere (e.g., National Mathematics Advisory Panel, 2008; National Council of Teachers of Mathematics [NCTM], 2000; U.S. Department of Education, 2003).
 Early mathematics intervention can repair deficits and prevent future deficits (Clements & Sarama, 2007; Fuchs, Fuchs, & Karns, 2001; Fuchs, Fuchs, Yazdian, & Powell, 2002; Griffin & Case, 1997; Sophian, 2004).
In mathematics, a reform process similar to what occurred in reading in the 1990s appears to be underway. Whereas math has been underresearched relative to reading, research findings are available to guide the application of RTI in mathematics. Specifically, research is available to guide the selection of adequate screening measures, selection of adequate progressmonitoring measures, development of decision criteria, and development of intervention protocols appropriate for use at all tiers of instruction. To use RTI in mathematics, a district or school must first select a model of RTI, identify adequate screening and progressmonitoring measures, and plan for effective delivery of intervention at Tiers 1, 2, and 3.
Useful Screening and Progress Monitoring Measures in Mathematics
Data are required for decision making in RTI. Generally, three decisions must be made: Who needs intervention? What type of intervention is needed? And is the intervention working?
Who Needs Mathematics Intervention?
To identify who needs intervention, educators need sensitive screening tools. The screening task should be a task that is closely aligned with expectations for learning in the classroom at that point in the instructional program. Published performance standards available in every state are an excellent basis for selecting a screening task. The screening task is used to make both relative (how one student’s performance compares to another’s in the same class or grade) and absolute judgments (how one student’s or all students’ performance compares to expectations for performance at that time in the year). Both judgments are necessary to correctly define a problem and determine what type of intervention or interventions are needed. Some screening models make only one judgment (relative or absolute), and this causes decision errors.
Curriculumbased measurement (CBM) probes of basic (e.g., sums to 12, subtraction 0–20, fact families) and advanced computation skills (e.g., finding least common denominator, multidigit multiplication with regrouping, converting numbers to percentages, solving equations) are empirically supported for screening (VanDerHeyden, Witt, & Naquin, 2003; VanDerHeyden & Witt, 2005). These measures have been found to yield reliable scores over time that correlate moderately with other more comprehensive measures of mathematics performance. Research indicates that the use of computationonly assessment and intervention has demonstrated value for early identification of children who are likely to struggle with advanced problem solving in mathematics. Because these probes can be administered to an entire class at one time and require only two minutes of the student's time, they are currently the measures of choice for screening in mathematics. To identify the screening task, the RTI consultant should print out the state standards for mathematics, review the computationoriented objectives in sequence, and consult with teachers at a gradelevel meeting to determine where students are in the instructional program (i.e., what students are expected to know how to do at that time of year to benefit from continued mathematics instruction). CBM probes can be purchased from a variety of sources (e.g., Sopris West Educational Services) or built for free using online tools (e.g., Intervention Central) or using inexpensive software (such as those on the Schoolhouse Technologies Web site).
What Type of Intervention Is Needed?
As noted previously, areas where many children (e.g., 25%–50% of those screened) perform poorly indicate the need for systemwide interventions in which all children receive the intervention (i.e., Tier 1). Where small numbers of children perform below the criterion (e.g., 2–4 students per class), small or largegroup intervention is indicated (i.e., Tier 2). Where only a few children perform poorly (e.g., fewer than 1 student per class on average), individualized intervention (i.e., Tier 3) may be immediately planned and implemented for those students. Alternatively, students who fail to respond at Tier 1 or Tier 2 may be provided with a higher level intervention—Tier 2 or Tier 3, respectively.
Following the collection of screening data, the decision team must determine whether a systemic problem exists. Where systemic learning problems are identified, the core program of instruction should be evaluated to ensure that a researchsupported curriculum is being used, that instruction is being delivered for sufficient duration and with sufficient quality, and that adequate resources are available to support effective instruction. The adequacy of the core instructional program in mathematics can be evaluated by comparing existing instructional procedures to elements of known effective instructional programs. Several panels have identified the use of routine assessment to continuously guide and refine instruction efforts (and effects) as a hallmark of effective instruction in mathematics (e.g., NCTM, 2000; U.S. Department of Education, 2003).
Effective mathematics instruction should include a system for monitoring student learning and adjusting instructional efforts to ensure adequate learning or accelerate it where needed. Other variables of effective instruction that are relevant include a wellsequenced program of instruction that logically builds on existing skills and periodically returns to previously mastered skills to ensure maintenance, demonstration of correct and incorrect responses, and substantial opportunity to practice performing newly learned skills with direct support (especially immediate corrective feedback) followed by more independent practice once the probability of errors is very low (or once accurate responding is a relatively sure thing).
Once a systemic problem is ruled out (or resolved through intervention), two groups of students might remain. First, there might be a subset of students who are performing below their classmates and in the risk range. These students’ performances may be similar to each other. These students may be targeted for Tier 2 intervention programming. Because their performances are similar, intervention materials and procedures can be geared toward the needs of the group (e.g., introduction of a skill to be learned, fluency building of an already acquired skill, teaching a prerequisite skill to fluency, guided practice to apply the skill under novel conditions). Tier 2 program features should include similar characteristics to those of effective Tier 1 programs (e.g., wellsequenced, ensuring mastery of skills as instruction progresses, adequate corrective feedback matched to student level of competence).
Effective Tier 2 programs for mathematics will emphasize matching the task difficulty to the capability of the students in the group, providing high numbers of opportunities to practice the skill and receive performance first under tightly controlled and stable conditions and later as the skill improves under variable conditions (e.g., with different materials, different problem presentations). Supplemental programs can be purchased to assist with intervention at Tier 2 (e.g., FASTT Math, Accelerated Math), and existing resources in the school can often be retooled to better serve the needs of children in need of Tier 2 intervention (e.g., the resource teacher or intervention coach could provide smallgroup intervention to Tier 2 students, with daily progress monitoring).
The second group likely to remain once Tier 2 students have been identified would comprise those students whose performances were below that of their classmates and in the risk range at screening, and for whom subsequent assessment shows extensive skill gaps. For these students, Tier 3 intervention should be implemented. Functional academic assessment will be necessary to build an intervention that adequately addresses weak skills for this group of students. Tier 3 programs should include a databased process for identifying specific causes of poor performance in mathematics, and individual interventions should be developed to target those specific deficits while monitoring both interventionspecific and generalized improvements in mathematics. Children who receive Tier 3 intervention may require specific training to learn how to apply learned skills under conditions that are required in the regular classroom.
Intervention procedures and materials at Tiers 2 and 3 might come from published resources (see the What Works Clearinghouse site for reviews of intervention programs), but upfront assessment will be needed to match the student with the right intervention and to obtain formative data to alter the intervention as needed to maximize intervention effects. Movement between the tiers can be bidirectional and children can go directly to the most intensive intervention level (Tier 3) if needed. Students who are already identified and receiving special education services could participate in RTI and receive instruction at Tiers 1, 2, or 3. Finally, assessment data collected at Tier 3 may be most useful for eligibility determination and individualized education program planning if a student is found eligible.
Is the Mathematics Intervention Working?
To evaluate intervention effects, two judgments must be made. First, was the intervention provided as planned for a sufficient period of time? Second, did student performance improve “enough?” The first question requires the collection of evidence of intervention implementation. The second question involves the collection of student performance data before intervention and with intervention relative to some criterion. To decide if intervention growth was sufficient, implementers might use a student performance criterion that reflects the average performance of students who are not at risk at screening at that school or that reflects the level of performance that has been associated with “passing” the accountability test in the spring, or the average performance of peers in the same class on that task. This second question poses greater measurement challenges in RTI (although research is making steady progress toward improving the adequacy of measurement tools and procedures for this purpose).
The most rigorous analysis of whether the intervention has successfully solved a learning problem is to consider postintervention performance on both the original and an updated screening task and evidence that the learning improvements caused by the intervention generalize to improved classroom performance and learning. Importantly, some researchers have demonstrated that direct measures of conceptual understanding in mathematics have a valueadded effect to understanding a student’s capacity in mathematics. In other words, computational skill is a meaningful and important precursor to successful mathematics learning, including applying learned skills to solve novel and more complex problems, but some students may demonstrate high levels of computational fluency with basic facts and operations yet fail to gain conceptual understanding. One logical and empirically supported approach is to address computation fluency problems where they are detected with intervention, and to directly assess conceptual understanding on related tasks (e.g., counting vs. more/less judgments; multiplication/division facts vs. solving a word problem that involves building equal sets).
Attending to Implementation Fidelity and System Change to Ensure Desired and Sustainable Outcomes for Mathematical Learning
It is important to note that whatever program of instruction is identified and implemented, research data tell us that deliberate planning and monitoring of implementation fidelity will be necessary to ensure desired outcomes. To enhance the sustainability of the RTI effort in mathematics, implementers should make every attempt to integrate the RTI math effort with ongoing system reform efforts. For example, RTI progressmonitoring data may have benefits in the following areas:
 Personnel review and professional development;
 Standardsbased instruction efforts;
 Efforts to promote effective instruction schoolwide, and
 Monitoring outcomes of change efforts
Implementation fidelity should be directly measured at all tiers. The most efficient way to monitor implementation fidelity is to track student performance (e.g., use the progressmonitoring data) and where student performance is not adequate, conduct a direct observation of instruction in the classroom to determine the percentage of intervention steps that are being completed as planned. This observation provides an occasion to provide direct coaching on how to effectively implement the intervention.
RTI is a logical system of databased decision making that can permit districts, schools, and teachers to evaluate the adequacy of ongoing mathematics instruction and to systematically chart a plan to accelerate learning in mathematics for all students and for those who are at risk for failure without intervention.
References
Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38, 136–163.
Fuchs, L. S., Fuchs, D., & Karns, K. (2001). Enhancing kindergartners’ mathematical development: Effects of peerassisted learning strategies. The Elementary School Journal, 101, 495–510.
Fuchs, L. S., Fuchs, D., Yazdian, L., & Powell, S. R. (2002). Enhancing first grade children’s mathematical development with peerassisted strategies. School Psychology Review, 31, 569–583.
Griffin, S., & Case, R. (1997). Rethinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 3, 1–49.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC.
Sophian, C. (2004). Mathematics for the future: Developing a Head Start curriculum to support mathematics learning. Early Childhood Research Quarterly, 19, 59–81.
U.S. Department of Education. (2003). Mathematics and science initiative concept paper. Retrieved June 1, 2004, from http://www.ed.gov/rschstat/research/progs/mathscience/concept_paper.pdf.
U.S. Department of Health and Human Services, Administration on Children, Youth and Families/Head Start Bureau. (2001). The Head Start path to positive child outcomes. Washington, DC: Author. Back To Top
