Get Email Updates
Sign up to receive the RTI Action Network enewsletter.

Implementing RTI in Mathematics 
June 19, 2008 1:00 PM  2:00 PM • Amanda VanDerHeyden, David AllsoppAbout this TalkJoin our experts Amanda VanDerHeyden and David Allsopp as they discuss "How can RTI play a role in helping students who are struggling in mathematics succeed?" while answering your questions about "Implementing RTI in Mathematics."
TranscriptSeymour S. Burack What specific diagnostic tests would you suggest to determine a true Math disability such as Dyscalculia, and how would the condition be treated. David AllsoppThere 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 studentâ€™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 curriculumbased measurement (CBM). Another resource that may be helpful is a document provided by the Center on Instruction titled Screening For Mathematics Difficulties in Grades K3 (Gersten, Clark, & Jordan, 2007) The document can be downloaded at www.centeroninstruction.org.
It is important to understand that the term â€œmath disabilitiesâ€ 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 visualspatial processing deficits and sequencing.
A multilens 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, problemsolving) and computation. Another good source for information on mathematics learning disabilities is on LDOnline.Â Vicki Peterman Are there math probes for problem solving?
Is there a need for probes beyond fluency for math? Amanda VanDerHeyden Yes. Intuitively we would all expect that assessment of computation and procedural fluency AND application or conceptual understanding would be important. particularly since so much of what we try to teach in math has to do with conceptual understanding (consider the stated goal of the National Mathematics Advisory Panel report just released that says the goal of primary math instruction is to get every child ready for algebra instruction by eighth grade).
Beyond intuition, however, are some pretty good data indicating that these are two related but distinct skills that merit separate measurement (Thurber & Shinn, 2002). Dr. Jim Connell at Temple University has completed a study replicating and extending the work of an earlier study completed by Lynn Fuchs and colleagues (1994). Typical correlations between computation probe scores (digits correct per minute) correlate with more comprehensive measures of mathematics performance in the range of r = .3 to .6. Connell developed a set of application measures or measures of conceptual understanding for primary grade students and adding these items to computation items improved the correlation from r = .6 to r = .8 with Iowa Test of Basic Skills scores. Importantly within RTI, however, is the idea that we are making different decisions at different stages. Some mathematics probes might be more efficient and more accurate for some types of decisions compared to other types of decisions (e.g., Connellâ€™s data suggested that computation probes were sufficient for screening decisions). Karen Decker Can you suggest some universal screeners in math, specifically for K5? Amanda VanDerHeydenSure, here are some ideas from my work. I suggest sitting down with a set of the state standards and identifying all the computation and procedure skills. This process alone is useful for teachers and administrators because often people are surprised to see an expected skill at a certain grade level and so this simple exercise prompts them to be sure that the instructional program is covering all that it should during the year. Given a sequence of expected skills, decision makers can identify those that will be most useful for three purposes

Determining whether systemlevel learning problems exist (i.e., many students are below expectations)

Identifying individual students who might need intervention and

Monitoring learning progress as the instructional program progresses.
With math, I always suggest two probes at each measurement occasion (a screening probe and a progress monitoring probe). If you start with a probe that represents a skill that students are expected to be able to perform at that point in the instructional program to continue to benefit from instruction that will be provided in math at their grade level, then you can determine whether there is a classwide, gradewide, or even systemwide learning problem in math (if a large percentage of students are below criterion). If a systemlevel learning problem is detected, it should be addressed first through Tier 1 intervention efforts (prior to identifying individual children for assistance). The reason for this is both logical (many children need help) and empirical (decisions about who is at risk are more accurate when the base rates are not high or stated another way lots of children are not below criterion).
If there is no systemwide learning problem, you can proceed with individual screening decisions. Decision makers would need to weigh the pros and cons of going the route of selecting an easier screening task/probe and administering that in the face of a system learning problem.
In addition to screening probe(s), a probe for progress monitoring is needed. The progress monitoring probe should represent a range of key skills that students are expected to master by some endpoint in the instructional program. So you could select a probe that allows you to monitor learning progress through the end of the first semester or through the entire school year. This means that early on, probe content would be difficult, but growth would be reflected as instruction is introduced. This approach to measurement is often tough for teachers and administrators to understand and they might ask, "why in the world would we assess a child on a skill for which they have not yet received instruction?" The answer to that is we want to make apples to apples comparisons to detect learning that occurs as instruction proceeds. So if the content of the measure changes then it will yield incomparable data. We use the same measure at all timepoints and as instruction progresses, performance will increase. The early measurement occasions can be thought of as baseline. Practically, with students, it only requires two (2) minutes of time to administer and students can be told that itâ€™s a challenging worksheet for which we will see growth during the year.Â Â
Â 
Â
Screening Fall

Â
Screening Spring

Â
Progress Monitoring

Â 1^{st} Grade 
Sums to 5 
Sums to 18 or 20 
Addition and Subtraction 020 
2^{nd} Grade 
Addition and Subtraction 020 
Multidigit addition or subtraction without regrouping 
Fact Families Addition/Subtraction 020 
3^{rd} Grade 
Fact Families Addition/Subtraction 020 or 3digit addition and subtraction with and without regrouping (this is hard for most third graders but reflects a skill that most are expected to be able to do) 
Multiplication 09 or 012

Multiplication and Division 012

4^{th} Grade
Â 
Fact Families Multiply/Divide 012
Â 
Multidigit multiplication without or with regrouping

Multidigit division with and without remainders

5^{th} Grade

Multidigit multiplication with and without regrouping 
1 digit into 23 digit dividend with remainders

Reduce fractions

6^{th} GradeÂ

Decimals multiplication

Find least common denominator

Substitution of whole number to solve equations 
Sue Keasler I teach preschoolers who are at risk. How will RTI affect and help me in a very hectic classroom w/ children who have a myriad of issues? What can I do for math readiness at this level? Amanda VanDerHeydenWe have something in common! I’ve been running research on early mathematics skill development with preschoolers at risk for a few years now. We built and tested a set of measures in 2004 (paper appeared in Journal of Early Intervention). They are timed CBM probes of early numeracy and assess skills like number identification, number naming, counting (onetoone object correspondence), free counting (count as high as you can), and visual discrimination (find the object/number/letter that is different). Others have built similar measures (e.g., Robin Hojnoski has a paper in School Psychology Review in 2006 describing a set of measures). Our measures are administered typically during the center play period.
The scores have been shown to be: (a) reliable over time, forms, and across scorers, (b) to correlate well with other measures of early skill development including the "Brigance Screens" and the "Test of Early Mathematics Ability," (c) to correlate with performance at kindergarten, and (d) to discriminate expected performance differences by grade level and risk status (this is some evidence of sensitivity).
We published the longitudinal data and the sensitivity data in a paper in Journal of School Psychology in 2006. What we have struggled with is demonstrating sensitivity of the measures for monitoring growth due to instruction and making short term (like week to week) decisions about altering instruction to improve learning (as you can do with CBM with older kids). We failed to demonstrate this type of sensitivity in both the 2004 and 2006 study, and others seem to struggle with this. too. Probably practitioners would not be surprised by this in thinking about working with typical preschoolers (they are variable and often unpredictable little creatures!). So based on our early failures, we designed and have now completed two followup studies. We are analyzing those data now, and I think we are on the right track. I have a set of kindergarten measures, too, and these are available through www.isteep.com. Linda Martinez As a parent, what can I do to encourage my child's school to implement RTI in mathematics to help him with his math problems? Amanda VanDerHeydenI can't help but wonder about the (likely) possibility that if your son is struggling then many children may be struggling in mathematics. When we conduct universal screenings in schools, we often find systemwide learning problems in mathematics. Often the deficits are apparent by end of year at grade 1 and just grow across the grades as the instructional program advances and children missing the prerequisite skills to benefit from that instruction.
If your school is using RTI for reading, then adding mathematics would not be too difficult. If RTI is new to your district, it might help to share some of the resources from this Web site with the district (www.RTINetwork.org). It also might help to take a look at the district data on the school or district Web site (most districts seem to have these data available now). You could probably detect from those data whether many children are struggling in the area of mathematics.
Ideally, you might find someone who is willing to screen even just one class. It only takes 25 minutes of time. A picture is worth a thousand words and people want to see their own data. When teachers select the probe, help administer/score and then see the graph showing child performance relative to expected performance, they are usually very enthusiastic to get on with the next step, whether it's classwide or individual intervention. Tracy Hurd How would this apply to high school students? David AllsoppYour 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 widescale 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 (concreterepresentationalabstract 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 decisionmaking. 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), 815.
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), 122. Andreia Ransdell I am involved in a team in my district that is currently designing documents to guide instruction in math at each grade level (k5) based on the TEKS. We are establishing Big Ideas, Units of Study, Essential Questions, Ideas for Differentiation, etc. Is there a way for us to integrate alreadycreated CBM tasks into that document? My goal is to connect teachers (in the easiest way possible) with CBM task resources already available, in the event that a student is not successful at the Tier 1 Level. To your knowledge, has this already been done? Is there a "bank" of math CBM tasks organized in such a way that would lend itself to being linked to the type of online instructional document my district is in the process of creating? Thank you in advance for your time and any input! David AllsoppThis 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 Fuchsâ€™ 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 districtâ€™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 â€“ University of Texas) has been working on an RTI tier 13 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. Â 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. Amanda VanDerHeydenWe ran into this in our work in a district in Vail, Arizona and ended up building our own. You can make probes using worksheet factory software. This will give you many of the probes you need/want. Sopris West sells their Basic Skill Builders product (Beck et al.) and in my opinion it's very good. I'm not affiliated with Sopris and don't receive any kickback or whatever. I just like their basic skill builders series for reading and for math. Interventioncentral.org might be another resource for you. Andreia Ransdell What are some examples of universal screeners commonly used in math at the elementary level? Amanda VanDerHeyden see earlier answer to karen decker Judy Jones How does RTI for math differ from RTI for reading? David AllsoppIn 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 13. 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 (concreterepresentationalabstract 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 decisionmaking.
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. Sally Gomez How do you offer tiered math instruction in a general ed classroom? This isn't just different skills levels  it's different content! David AllsoppThree 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 onetoone situations, occur in the general education classroom. It is likely that this would happen in a coteach 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 (pullout 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 ongrade 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 coteach situations. While they would both be tier 1, the coteach 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 13. 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 (concreterepresentationalabstract 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 decisionmaking. 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 Kelly If your scores are not quite at the range where you can implement RTI (yet!), what strategies or schoolwide interventions would you recommend? David AllsoppI 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 schoolwide intervention for the purpose of preventing academic difficulties including unnecessary/incorrect identification of disability. At the schoolwide level, the research base supports the use of a number of math instructional practices that can be applied to any K12 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 (concreterepresentationalabstract 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 onetoone situations.
Here are some resources that you might find helpful for more information about these practices as well as others:
You may also find the work being done by Bryant & Bryant at the University of Texas on tier 13 interventions helpful:
Hope this helps! Amanda VanDerHeydenI'm not sure I understand about not being able to use RTI. RTI procedures (interventions, progress monitoring) are a great way to improve skills and raise scores! You could start with 15 minutes per day of classwide intervention in classes where scores are below what you would like them to be. With periodic probes of classwide performance, you can evaluate whether the class is showing growth and getting on track.
We have some intervention protocols up on www.gosbr.net. The class is divided into working pairs and the protocol provides guided practice with immediate corrective feedback, independent timed practice with a goal to beat, error correction, and rewards for beating the last best score. These intervention components come mostly from studies on behavioral change in education (applied behavior analysis, direct instruction, precision teaching). Susan Davis Are there core components essential to mathematics skill development, similar to the five core components of reading? David AllsoppThis 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 K12 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.Â The title of the article is Number Sense: Rethinking Arithmetic Instruction for students with Mathematical Disabilities.
Additional Resources on RTINetwork.org

