The Center on Accelerating Student Learning (CASL) is a collaborative partnership among faculty at Vanderbilt University (Lynn and Doug Fuchs), Columbia University (Joanna Williams), and the University of Maryland (Steve Graham and Karen Harris, Vanderbilt University). CASL's goal is to identify instructional practices that accelerate the learning of children with disabilities in kindergarten through Grade 3. This includes the development of effective, multicomponent instructional interventions in reading, writing, and math, which focus on basic skills and higher-order learning, and promote fluency, transfer, and maintenance. As part of the CASL research program, we validated an intervention to enhance mathematical problem-solving at third grade. We call this intervention "Hot Math." (See Fuchs, Fuchs, Prentice, Burch, & Paulsen, 2002.)
More recently, within the context of another grant sponsored by the U.S. Department of Education's Office of Special Education Programs, we have been studying how hot Math might be used within a multitiered response-tointervention (RTI) system. The first intervention tier occurs in general education. The second tier involves more intensive tutoring of children deemed at risk for poor problem-solving outcomes, even with the validated hot Math Tier 1 intervention. In other areas, primarily reading at first grade, RTI has undergone some study (e.g., Speece & case, 2001; Vellutino et al., 1996). In math, where less research has been conducted, the only major experimental RTI study was conducted at first grade, addressing more foundational skills (Fuchs et al., 2005b). Our goal was to extend RTI to third-grade math problem-solving.
Toward that end, across 2 years, we worked in 13 schools. At Tier 1, we randomly assigned 20 general education classrooms to conventional methods for teaching math problem-solving and the other 40 general education classrooms to Tier 1 hot Math whole-class instruction, that occurred 2 to 3 times per week for 16 weeks for 25 to 40 minutes per session. We assessed students in these 60 classrooms at the beginning of the school year to identify children for Tier 2 intervention, and randomly assigned two thirds of the 144 lowest-scoring students to Tier 2 hot Math tutoring and one third to no tutoring. Tier 2 hot Math tutoring occurred three times weekly for 13 weeks. We could estimate the effects of each tier of hot Math intervention separately and in combination because students had been randomly assigned to four conditions: (a) no hot Math in either Tier, (b) hot Math only at Tier 1, (c) hot Math only at Tier 2, and (d) hot Math at both tiers.
How We Structured Intervention at Tiers 1 and 2
Tier 1 hot Math Whole-Class Instruction
Hot Math integrates two practices to promote mathematical problem-solving:
* Explicit instruction about transfer.
* Self-regulation strategies.
The first 3-week unit is dedicated to basic problem-solving information: making sure answers make sense, lining up numbers from text correctly to perform math operations, and labeling work with words and mathematics symbols. Each of the next four 3-week units focuses on one of these problem types (see Table 1; also see Fuchs et al., 2002 for sample problems):
* "Buying Bags" problems.
* "Shopping List" problems.
* "Half" problems.
* "Pictograph" problems.
Teachers can apply the instructional principles to other problem types as well.
Each of the four units provides instruction on skill acquisition and transfer. In each unit, the first four sessions focus on skill acquisition (what the problem is asking and how to solve the problem), with cumulative review across units incorporated in Session 4. During the third week of each unit, teachers provide two transfer lessons in Sessions 5 and 6, incorporating cumulative review again in Session 6. In each unit, the first acquisition lesson (Session 1) and the first transfer lesson (Session S) last 40 minutes; the others last 25 to 30 minutes.
To teach rules for problem solution, Tier 1 structures all problems in the same way, but cover stories and quantities vary. A poster listing the steps of the solution method is displayed. In Session 1 of each 3-week unit, teachers present a worked example and, as they refer to the poster, explain each step of the solution method. After presenting several worked examples in this way, teachers move to partially worked examples, where students work in pairs to apply steps of the solution and report answers. Eventually, in each lesson, students complete several problems entirely in pairs, with stronger students helping weaker students. Each lesson ends with students completing one problem independently, with the teacher checking answers against a scoring key. Finally, teachers assign students a homework problem, that they return the next morning to the homework collector (a competent student in the class) for checking. The program structures Sessions 2 to 4 of each unit similarly to Session 1 except that Sessions 2 to 4 allocate more time to dyadic practice than to directed instruction.
In Sessions 5 to 6, teachers address transfer in two ways. First, teachers explicitly teach the concept of transfer. Then, they present four types of problem features that can change a problem without altering its structure or solution: A familiar problem type, which the child knows how to solve, can be formatted so that the problem looks novel; can use an unfamiliar key word; can pose an additional question; can combine problem types; can present irrelevant information. A poster lists the four "Ways Problems Can Change."
In Session 5, teachers explain the poster. Then, students classify problems (of that unit's problem structure) according to which problem feature has changed and explain how problems seem different but still represent the problem type. Teachers next have students work in pairs to solve transfer problems, which systematically vary one feature at a time. Students complete and check one transfer problem independently, and teachers assign homework. Session 6 is structured similarly to Session 5, except that it also incorporates cumulative review across units.
As reflected in the content of the transfer lessons, mathematical problem-solving requires metacognition, whereby children regulate the selection and use of skills and strategies. For this reason, we also incorporate self-regulated learning strategies of goal setting (to help motivate children to focus their effort and work hard) and self-monitoring (to facilitate goal attainment). These self-regulated learning strategies are incorporated into each session with three additional activities as follows:
1. Students get feedback on their performance on each session's final, independent problem.
2. Students graph these daily scores on their personal thermometer chart, which shows consecutive thermometers (one for each session) with each thermometer going from O to the maximum score for that problem type.
3. At the beginning of the next session, students inspect their charts and set a goal to beat their highest score on the day's independent problem.
All of the Tier 1 whole-class sessions are scripted to help teachers master the instructional format and content.
Tier 2 hot Math Tutoring
Hot Math tutoring occurs in groups of 2 to 4 students, three times per week for 13 weeks, each time for 20 to 30 minutes. The instruction incorporates a strong self-regulated learning component with tangible reinforcers, which we have found to be important in effecting learning outcomes. Students earn "dollars" for on-task behavior and for accurate work products; at the end of each week, they can spend their dollars on trinkets from the hot Math store.
The content of hot Math tutoring mirrors the material covered in the whole-class sessions, except that more difficult concepts are targeted for instruction, with additional use of concrete manipulatives and extra prompts to support student learning. see Table 1 for information for one unit (Buying Bags) on how the content of the Tier 2 tutoring aligns with, supports, and supplements the material covered in Tier 1 whole-class tutoring. (All of the Tier 2 tutoring sessions are scripted.)
What We Found
We examined improvement from the beginning to the end of third grade on three mathematical problem-solving measures. The students had never before seen any of the problems on any of the measures.
* The first measure, which we call "immediate transfer," asks students to complete problems that are structured like the problems used for teaching the problem type, but with novel cover stories.
* The second measure, which we call "near transfer," asks students to complete problems that include novel cover stories as well as one or more novel transfer features.
* The third measure, which we call "far transfer," asks students to answer four questions about a problem narrative, which requires students to use all four problem types addressed during hot Math (as well as six additional math skills embedded in the typical third-grade curriculum) and calls for students to apply all transfer features taught during hot Math.
We found evidence for intervention efficacy on the first two types of measures. On the far-transfer test, which mirrored complex, real-life problemsolving situations, results were in the direction favoring hot Math efficacy, but did not reach statistical significance.
In addition, in terms of RTI, we designated "lack of responsiveness" as scoring more than a standard deviation below the growth of a normative sample (close to 600 students) who received Tier 1 whole-class hot Math intervention. This means that any student whose growth in response to hot Math was below the 16th percentile was deemed unresponsive to the program. This notion of unresponsive is central to RTI, which targets children for special education whose growth is considered unacceptably low.
Using this cut-point for growth as a way of defining lack of response, we documented rates of unresponsiveness as a function of the tiers of hot Math intervention students received. On the immediate- and near-transfer measures, respectively, 86% and 100% of control students (who did not receive Tier 1 or Tier 2 hot Math) were unresponsive in math problem-solving when they received the conventional math problem-solving instruction designed by their teachers. This high rate of failure is disconcerting, and it is reassuring that with at least one tier of validated intervention with hot Math, the rate of failure decreased substantially: among students who received Tier 1 hot Math whole-class instruction but without Tier 2 hot Math tutoring, to 29% (vs. 86%) on immediate transfer and to 54% (vs. 100%) on near transfer; among students who received Tier 2 hot Math tutoring but without Tier 1 hot Math whole-class instruction, to 55% (vs. 86%) on immediate transfer and to 62% (vs. 100%) on near transfer.
Rates of unresponsiveness were, however, most impressive when at-risk students received both tiers of hot Math intervention: 12% on immediate transfer and 26% on near transfer. These findings illustrate how two tiers of intervention, designed strategically to work in supplementary and coordinated fashion, may operate synergistically to decrease math problem-solving difficulties for children who are otherwise at risk for poor outcomes.
Conclusions must be tempered by the absence of follow-up data to determine how well effects persist over time. Pending long-term data, however, we tentatively conclude that two tiers of validated, coordinated math problemsolving intervention can substantially reduce third-grade risk for poor math problem-solving.
Our controlled experimentation also indicates that RTI, which incorporates validated hot Math intervention at Tiers 1 and 2, may represent a promising structure for preventing and identifying math disability relatively early in a child's school experience. Identifying nonresponders for math problemsolving in this way, by the end of third grade, permits highly skilled special educators to intervene in math problemsolving more intensively and earlier than typically occurs, with the goal of enhancing long-term outcomes for students with math learning disabilities.
Classroom Teacher Perspectives
Ms. West is a third-grade teacher in the Metropolitan Nashville Public Schools:
This is the second year that my classes have participated in the hot Math program. Teaching problem-solving skills has always been a bit difficult. However, with hot Math, my students are much better word-problem solvers because they have a foundation for further classroom instruction. I was very pleased with the gains my students made last year from beginning of unit testing to end of unit testing. It was amazing to see how most of them had grown. The partner work allows students to "teach" one another and to practice skills taught in the lesson. I love to listen to my students as they make explanations to one another. I also love the systematic, regimented approach that calls for consistent review of previously taught material. hot Math is an excellent program. I certainly hope it is offered again for the 2006-07 school year. I definitely want to participate again!
Mrs. Tubb is a third-grade teacher in the Metropolitan Nashville Public Schools:
The Vanderbilt hot Math project is an engaging, fun, creative approach to help unlock the "mystery" of problem-solving for young children in third grade. Units are presented that focus on targeted expectations that allow children to experience success. A variety of strategies allow participation and cooperation in the class setting. Each child has a partner with whom he/she discusses ideas and solves certain tasks as they are presented for partner-work. Progressing toward more independence, as understanding is achieved, an opportunity is provided that allows for each child to solve a problem alone. A teacher observes and scores the steps each child followed. Before the class ends, "thermometers" have been colored in to show how "hot" the child was that day in math. As the sessions pass, many children become aware of what they need to do to solve word problems correctly. It is time well-spent during the weeks of the project. It appeals to the self-assured math "whizzes," as well as the reluctant learners. Everyone is involved and looks forward to those special sessions. "hot Math" is an effective tool to incorporate into the instructional setting. The more involved children are in the process, the more they take away and apply later on as K needed. My class truly enjoyed being a part of this project.
Ms. Graf is a third-grade teacher in the Metropolitan Nashville Public Schools:
It is so nice for the students to have extra 'professional' help in math. Small group and one-on-one tutoring make such a difference in building math skills!
Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J., & Hamlett, C. L. (2005a). Responsiveness to intervention: Preventing and identifying mathematics disability. TEACHING Exceptional Children, 37(4), 60-63.
Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005b). The prevention, identification, and cognitive determinants of math difficulty. Journal of Educational Psychology, 97, 493-513.
Fuchs, L. S., Fuchs, D., Prentice, K., Burch, M., & Paulsen, K. (2002). hot Math: Promoting mathematical problem solving among third-grade students with disabilities. TEACHING Exceptional Children, 31(1), 70-73.
Speece, D. L., & case, L. P. (2001). Classification in context: An alternative approach to identifying early reading disability. Journal of Educational Psychology, 93, 735-749.
Vellutino, F., Scanlon, D. M., Sipay, E. R., Small, S. G., Pratt, A., & Chen, R. (1996). Intervention as a vehicle for distinguishing between cognitive and experiential deficits as basic cause of specific reading disability. Journal of Educational Psychology, 88, 601-838.
Address correspondence to Lynn S. Fuchs, 328 Peabody, Vanderbilt University, Nashville, TN 37203 (e-mail: email@example.com). For manual with scripts for Tier 1 and Tier 2 instruction, contact flora.murray@ vanderbilt.edu
Research described in this article was supported by Grants #H324C030115 and H324V980001 from the U.S. Department of Education, Office of Special Education Programs, and Core Grant HD15052 from the National Institute of Child Health and Human Development to Vanderbilt University. Statements do not reflect the positions or policies of these agencies, and no official endorsement by them should be inferred.…