Implementing RTI in Mathematics

David Allsopp
There has been quite a lot of discussion about what math learning disabilities are, how they are similar and dissimilar to other learning disabilities such as ones that affect reading, and how to assess their occurrence. Traditionally, a discrepancy model has been used to determine learning disabilities in mathematics as well as other areas of learning. This typically involves looking for a significant difference between a student’s IQ score and one or more scores on subtests of a mathematics achievement battery. Additionally, some have also examined significant discrepancies among achievement and cognitive processing batteries; that is, looking for any obvious inconsistencies among particular areas of mathematical abilities (calculation, application) and cognitive processing (e.g., visual, auditory, motor, processing speed, etc.). The validity of this approach has been called into question in recent years. Some researchers suggest that students with math learning disabilities have common areas of difficulty including difficulty with numeracy/number sense , slow processing speed, and difficulty with working memory. Recently, RTI has been suggested as a possible alternative to identifying mathematics learning disabilities. Fuchs, Compton, Fuchs, Paulsen, Bryant, & Hamlett (2005) suggest its use as a process for identifying mathematics disabilities at the end of the 1st grade year. They found that the best predictors for identification were 1) low performance on end of the year mathematics achievement tests that assesses 1st grade concept application and computation; and 2) poor rate of growth across the year using curriculum-based measurement (CBM). Another resource that may be helpful is a document provided by the Center on Instruction titled Screening For Mathematics Difficulties in Grades K-3 (Gersten, Clark, & Jordan, 2007) The document can be downloaded at www.centeroninstruction.org. It is important to understand that the term “math disabilities” does not refer to one type of difficulty. Mathematics learning disabilities are multifaceted and can be the result of a variety of problematic areas of learning. In addition to those areas already mentioned, language difficulties can be an issue, as can be visual-spatial processing deficits and sequencing. A multi-lens perspective is likely the best approach at this point in terms of assessment for mathematics learning disabilities. One might want to place special emphasis on areas such as numeracy/number sense, processing speed, working memory, and combined performance on end of year achievement test and yearlong growth demonstrated via CBM. Any assessment should include both conceptual application (i.e., reasoning, representation, communication, connections, problem-solving) and computation. Another good source for information on mathematics learning disabilities is on LDOnline. 

David Allsopp

Your question is an important one. Unfortunately, it is also one where clear answers are not readily available. Mathematics for RTI for schools generally is not nearly as advanced as it is, say, for reading. This is much less the case for high school. Compounding the issue at the secondary level is that mathematics becomes much more specialized in terms of content. Additionally, graduation requirements and state assessments are tied to particular mathematics courses (e.g., Algebra 1, Geometry 1, etc.). I am not aware of any wide-scale application of mathematics and RTI at the high school level. This is not to say that RTI is not happening at some high schools. However, in terms of systematically applied and evaluated models, there are none that I know of at this time.

The good news is that there is a research base from which secondary educators can base their tiered instruction. There are a number of mathematics instructional practices that appear to be promising from a research perspective, all of which can be applied to secondary settings. They include 1) explicit systematic instruction within authentic contexts; 2) teaching strategies for learning and doing mathematics including use of graphic organizers; 3) grounding abstract concepts within concrete experiences (concrete-representational-abstract sequence of instruction); 4) providing multiple opportunities for students to apply their mathematical understandings (both newly learned concepts and those for maintenance); and 5) continuous progress monitoring/instructional decision-making. For more information and video models of these practices and others, go to the MathVIDS website: http://coe.jmu.edu/mathvids2 For a recent synthesis of research on effective mathematics instruction for students with mathematics difficulties, see Gersten, Baker, & Chard (2006). Effective Instructional Practices for Students with Difficulties in Mathematics: Findings from a Research Synthesis. Center on Instruction, www.centeroninstruction.org .

Several articles that address effective instruction for struggling learners/disabilities are:

Gagnon & Maccini (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 34(1), 8-15. Maccini & Gagnon (2000). Best practices for teaching mathematics to secondary students with special needs: Implications from teacher perceptions and a review of the literature. Focus on Exceptional Children, 32(5), 1-22.

David Allsopp
This is an excellent question and one that I hope elicits more action on the part of researchers. To date the vast majority of probes that are available are computation driven. One exception to this is the CBM work being done at Vanderbilt University. The Fuchs’ and their colleagues have developed a series of probes that include computation and application type items. You might want to examine examples that are posted online to determine the extent to which they are aligned, or could be aligned, with your district’s curriculum. A summary of their work with examples of probes are provided in Using Curriculum Based Measurement for Progress Monitoring in Math (Ideas That Work, US Office of Special Education Programs & studentprogress.org). Another group (Bryant & Bryant – University of Texas) has been working on an RTI tier 1-3 project and may also have some probes that might go beyond only computation type tasks. They have been piloting a tiered mathematics intervention for early grades mathematics. More information about this project can be found at www.texasreading.org.  I believe they conducted a webinar through the Access Center where a copy of the presentation can likely be downloaded. Another resource that provides examples of mathematics probes is Aimsweb.

David Allsopp

In concept, RTI for mathematics does not differ from RTI for reading. The premise is the same - to provide all students with effective instruction that is supported by research in order to prevent school failure or the need for identification for special education services. In terms of process, there would be little difference. The structure (e.g., screening, tiered instruction, use of continuous progress monitoring) will not differ. What will be different is the type of instructional practices that will be implemented and the concepts and skills that will be emphasized. The fact is that the application of RTI for mathematics is in its infancy compared to reading and behavior. Therefore, there is much less to go on in terms of models at this point. Bryant and Bryant at the University of Texas are doing some initial research into a tiered mathematics intervention. You might find their work interesting. They describe a model for intervention at tiers 1-3. You can download a presentation of their work at the Vaughn Gross Center for Reading and Language Arts (VGC) Web site. They also completed a webinar with The Acccess Center. There are a number of mathematics instructional practices that appear to be promising from a research perspective.

They include:

Explicit systematic instruction within authentic contexts;

Teaching strategies for learning and doing mathematics including use of graphic organizers;

Grounding abstract concepts within concrete experiences (concrete-representational-abstract sequence of instruction);

Providing multiple opportunities for students to apply their mathematical understandings (both newly learned concepts and those for maintenance);

Continuous progress monitoring/instructional decision-making.

For more information and video models of these practices and others go to the MathVIDS Web site. For a recent synthesis of research on effective mathematics instruction for students with mathematics difficulties see Gersten, Baker, & Chard (2006). "Effective Instructional Practices for Students with Difficulties in Mathematics: Findings from a Research Synthesis." Center on Instruction. www.centeroninstruction.org.

David Allsopp

Three key principles of tiered instruction as it relates to the general education curriculum are 1) to ensure that all students receive effective instruction based on sound research, 2) to provide increasing levels of support and explicitness as students demonstrate difficulties, 3) to provide opportunities for students to move back and forth among the tiers based on their needs.

How support is provided for those students in tiers 2 and 3 is dependent upon how one’s school or district is implementing RTI for mathematics. In some cases, tier 2 and 3 interventions, which typically occur in small group or one-to-one situations, occur in the general education classroom. It is likely that this would happen in a co-teach situation or with a teacher who is very experienced with implementing differentiated group instruction. In other cases, tier 2 and 3 interventions might occur in other contexts with a specialist (pull-out or pull in with math coach).

It is true that some students will need additional work with “below grade level” concepts which may be what you are referring to in your question when you say, “its different content.” One determining factor for the tier a particular student will receive mathematics instruction is the extent to which gaps in mathematical knowledge affect her/his ability to master on-grade level mathematics. Some students may need additional support (e.g., tier 2) for certain below grade level concepts/skills but may have a solid enough base to receive most of their instruction at tier 1. For others, this might not be the case. They may receive all of their mathematics instruction at tier 2 or 3. The hope would be that they would progress to the point where over time, they would have a solid enough base to move “down” one or more tiers.

One thought is that schools could scaffold support within the general education classroom within a given tier. For example, in tier 1 perhaps some classrooms would have one teacher who is experienced with differentiated instruction. Other classrooms would be co-teach situations. While they would both be tier 1, the co-teach classroom would be made available for those students who need slightly more support than other students who are in tier 1. By doing this, there is more flexibility for teachers to emphasize what the do best and for students to receive the type of scaffolded support they require within a given tier. Perhaps such an arrangement would assist teachers in working with students who need remedial work due to gaps in mathematical knowledge.

The fact is that the application of RTI for mathematics is in its infancy compared to reading and behavior. Therefore, there is much less to go on in terms of models at this point. Bryant and Bryant at the University of Texas are doing some initial research into a tiered mathematics intervention. You might find their work interesting. They describe a model for intervention at tiers 1-3. You can download a presentation of their work at the Vaughn Gross Center for Reading and Language Arts (VGC) Web site. They also completed a webinar with The Acccess Center that is available on the Access Center's site.

There are a number of mathematics instructional practices that appear to be promising from a research perspective. They include 1) explicit systematic instruction within authentic contexts; 2) teaching strategies for learning and doing mathematics including use of graphic organizers; 3) grounding abstract concepts within concrete experiences (concrete-representational-abstract sequence of instruction); 4) providing multiple opportunities for students to apply their mathematical understandings (both newly learned concepts and those for maintenance); 5) continuous progress monitoring/instructional decision-making. For more information and video models of these practices and others’ go to the MathVIDS website: http://coe.jmu.edu/mathvids2

For a recent synthesis of research on effective mathematics instruction for students with mathematics difficulties see: Gersten, Baker, & Chard (2006). Effective Instructional Practices for Students with Difficulties in Mathematics: Findings from a Research Synthesis. Center on Instruction, www.centeroninstruction.org

David Allsopp

I hope that I understand your question correctly. It sounds as if you are referring to what can be done at Tier 1 and Tier 2 levels prior to testing/identification for special education services. I hope I am on the right track.

Actually, RTI is about school-wide intervention for the purpose of preventing academic difficulties including unnecessary/incorrect identification of disability. At the school-wide level, the research base supports the use of a number of math instructional practices that can be applied to any K-12 math curriculum. They include: 1) explicit systematic instruction within authentic contexts; 2) teaching strategies for learning and doing mathematics; 3) grounding abstract concepts in concrete experiences (concrete-representational-abstract sequence of instruction); 4)providing multiple opportunities for students to apply newly acquired math knowledge (as well as for maintenance of previously learned mathematics); and 5)continuous monitoring of progress to make timely instructional decisions. These practices can be applied in the general education classroom in whole class, small group, and one-to-one situations.

Here are some resources that you might find helpful for more information about these practices as well as others:

• Center on Instruction
• MathVIDS
• Special Connections 

You may also find the work being done by Bryant & Bryant at the University of Texas on tier 1-3 interventions helpful:

Hope this helps!

David Allsopp

This is an interesting question! Actually, there seems to be some correlation between what has been learned in reading and what it may mean for mathematics. An area that seems to be most critical for K-12 mathematics success is the area of number/number sense. For example, lack of number sense seems to be a consistent issue for students who fail Algebra at the secondary level.

Along with this, I personally believe that students, particularly those that are struggling to learn mathematics, be exposed to multiple processes of doing mathematics. The National Council of Teachers of Mathematics (NCTM) suggests five processes for doing mathematics: reasoning/proof, representation, communication, connections, problem solving. Too often, students who struggle get exposed to only procedural mathematics (i.e., computation) while some of these other processes for doing mathematics could actually assist with strengthening their conceptual understandings.

As it relates to RTI, one might want to place an emphasis on number sense across these different processes for students who are at risk or who demonstrate difficulties. This could also provide a sound base for a school's screening battery.

For more information on the processes for doing mathematics you can go to the NCTM web site.

Also, Russell Gersten and his colleagues wrote an interesting piece on this topic. You can find it at ldonline.  The title of the article is Number Sense: Rethinking Arithmetic Instruction for students with Mathematical Disabilities.