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ff8)d,z9:ssssN40+++++++$.01t+LNNLL+ss+fffLpss+fL+ffV!@!s0kdRU!++0,a!01R1!!1'LLfLLLLL++XLLL,LLLL1LLLLLLLLLf o: Elements of Effective Mathematics Interventions
TopicEstablishedEmergingNot in placeContentIs the content of the intervention appropriate for accelerating each childs academic growth? Do the instructional activities contain challenging tasks that are appropriate for studentsinterests and backgrounds?
Intervention materials for students receiving interventions should focus intensely on indepth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee. The specific content of the interventions is centered on building the students foundational proficiencies. Fewer topics are covered in more depth, and with coherence.
Students who are not fluent with basic math facts should receive about 10 minutes in each intervention session to building fluent retrieval. Students are taught or develop strategies to aid in fluency, including use of number properties.Intervention materials address basic procedures related to computation but not the underlying concepts, and dont provide significant problem solving opportunities related to computational fluency.Intervention materials cover topics that are not essential to building basic competencies, such as data analysis, measurement and time. (While these are important topics in the grade level curriculum, the interventions should focus on foundational concepts and skills that are used in contexts such as data analysis, measurement and time.)Do the instructional activities advance the schools curriculum by promoting reasoning,conceptual understanding and problem solving?Interventions should include instruction on solving word problems that is based on common underlying structures. Students are taught about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solution strategies for each problem type. Students are taught to recognize the common underlying structure between familiar and unfamiliar problems (based on concepts) and to transfer known solution methods from familiar to unfamiliar problems.Curriculum materials classify problems into problem types, but students do not spend much time learning how to solve various types nor do they gain insight into the deeper mathematical ideas in the word problems. Also, there is little effort to attend to more difficult problem types.
Curriculum materials do not classify problems into problem types, do not focus on the mathematical ideas in word problems, and do not teach about connections between various types of problems.
Is the mathematics in the instructional activities correct?Yes, both conceptually and procedurally, and it is adequately represented visually.Yes, although it is not adequately represented visually.No; or interventionists are not expert with the underlying mathematics content.CurriculumDo the instructional activities used in interventions support and enhance, but not supplant or duplicate, regular classroom instruction?Intervention classes do not replace regular classroom instruction. Activities used during intervention support and enhance regular classroom instruction and students see the connection to their general classroom instruction. Activities used during intervention are related to regular instruction, but the connections are not made explicit to the student.Instructional activities used during intervention are not related to regular classroom instruction in any way; or students do not receive regular classroom instruction, only supplemental.Does the intervention program include a series of instructional activities that are carefully linked with the diagnostic assessments?Diagnostic assessments are used to determine areas of need for each student. Instructional activities are carefully linked with the diagnostic assessment results.Some informal diagnostic assessments are made when students enter interventions.The content of interventions is the same for all students regardless of their specific needs.InstructionIs instruction explicit and systematic as needed?
HYPERLINK "http://dww.ed.gov/profiles/?T_ID=28&P_ID=71&sID=310"
Organizing for Differentiation in the Core Classroom
Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. Instructional materials include numerous clear models of easy and difficult problems, with accompanying teacher thinkalouds. Students are provided with opportunities to solve problems in a group and communicate problemsolving strategies. Instructional materials include cumulative review in each session.Interventionist attempts explicit instruction in a way that is not systematic. Intervention does not include cumulative review in each session.
Interventionists do not know how to provide explicit instruction. Intervention materials do not incorporate sufficient models, thinkalouds, practice or cumulative review.
Does instruction include use of visual representations to model abstract mathematical concepts and processes?
HYPERLINK "http://dww.ed.gov/profiles/?T_ID=28&P_ID=71&sID=318" Concrete to Abstract Sequence
Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. Students learn to express mathematical ideas using visual representations such as number lines, arrays, and strip diagrams and to convert visual representations into symbols used in problem solving. When visual representations are not sufficient, concrete manipulatives are used.Students are exposed to visual representations only occasionally and unsystematically. Little effort is made to translate visual representations into symbolic procedures.Classroom tasks focus on procedural manipulation without being tied to visual representations. Teachers may believe that the use of visual representations and concrete manipulatives take too much time. Teachers may not understand the mathematical ideas that underlie some of the representations.Are tools for ongoing, formative assessment embedded in the instructional activities?
HYPERLINK "http://dww.ed.gov/see/?T_ID=28&P_ID=71"Explicit Teaching in the FifthGrade Math CoretTeachers use formative assessment processes to provide ongoing descriptive feedback to students and to modify instruction as needed to promote learning.Occasional use of formative assessment occurs in interventions, but is not planned and results are not used systematically.Teachers are not using formative assessment on an ongoing basis and do not have information to say which students are not understanding the concepts prior to the summative assessment.Are districts systematically using
universal screening to determine which students have mathematics difficulties and require researchbased interventions?Districts select screening measures based on the content they cover, with an emphasis on critical instructional objectives for each grade. They screen all students in the beginning and middle of the year. Because screening is just a means of determining which students need help, more time is invested in diagnostic assessment of students who perform poorly on the universal screening measure.Screening assessments cover the full range of GLCEs rather than focusing on critical foundational objectives for each grade, or schools use the results of large scale assessments (like MEAP) to place students in interventions.Students are placed in interventions based on casual observation rather than through data collection and the problemsolving approach.Are districts using progress monitoring assessments to determine which students are learning from the intervention and which may need more intense interventions?Interventionists use curriculumembedded assessments in interventions to determine whether students are learning from the intervention. These measures can be used as often as every day or as infrequently as once every other week. Students are released from intervention support or recommended for more intensive support as indicated by the results of progress monitoring assessments.Progress monitoring assessments are informal and lack reliability and the ability to track growth.No system is in place to track growth over time and make decisions related to placement of students in various levels of support.
Additional explanation from the IES Practice Guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools:
For most students, the content of interventions will include foundational concepts and skills introduced earlier in the students career but not fully understood and mastered. Whenever possible, links should be made between foundational mathematical concepts in the intervention and gradelevel material.
The panel focused on practices that addressed the following areas of mathematics proficiency: operations (either computation or estimation), concepts (knowledge of properties of operations, concepts involving rational numbers, prealgebra concepts), problem solving (word problems), and measures of general mathematics achievement. Measures of fact fluency were also included because quick retrieval of basic arithmetic facts is essential for success in mathematics and a persistent problem for students with difficulties in mathematics.
The panel believes that districts should review the interventions they are considering to ensure that they cover concepts involving rational numbers in depth. The focus on rational numbers should include understanding the meaning of fractions, decimals, ratios, and percents, using visual representations (including placing fractions and decimals on number lines see recommendation 5), and solving problems with fractions, decimals, ratios, and percents.
The documents reviewed demonstrate a growing professional consensus that coverage of fewer mathematics topics in more depth and with coherence is important for all students. Milgram and Wu (2005) suggested that an intervention curriculum for atrisk students should not be oversimplified and that indepth coverage of key topics and concepts involving whole numbers and then rational numbers is critical for future success in mathematics. The National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points (2006) called for the end of brief ventures into many topics in the course of a school year and also suggested heavy emphasis on instruction in whole numbers and rational numbers.
The National Mathematics Advisory Panel defines explicit instruction as follows (2008, p. 23):
Teachers provide clear models for solving a problem type using an array of examples.
Students receive extensive practice in use of newly learned strategies and skills.
Students are provided with opportunities to think aloud (i.e., talk through the decisions they make and the steps they take).
Students are provided with extensive feedback.
The NMAP notes that this does not mean that all mathematics instruction should be explicit. But it does recommend that struggling students receive some explicit instruction regularly and that some of the explicit instruction ensure that students possess the foundational skills and conceptual knowledge necessary for understanding their gradelevel mathematics. Our panel supports this recommendation and believes that districts and schools should select materials for interventions that reflect this orientation. In addition, professional development for interventionists should contain guidance on these components of explicit instruction.
Explicit instruction typically begins with a clear unambiguous exposition of concepts and stepbystep models of how to perform operations and reasons for the procedures. Interventionists should think aloud (make their thinking processes public) as they model each step of the process. They should not only tell students about the steps and procedures they are performing, but also allude to the reasoning behind them (link to the underlying mathematics).
The panel suggests that districts select instructional materials that provide interventionists with sample thinkalouds or possible scenarios for explaining concepts and working through operations. A criterion for selecting intervention curricula materials should be whether or not they provide materials that help interventionists model or think through difficult and easy examples.
The panel believes that content covered in a screening measure should reflect the instructional objectives for a students grade level, with an emphasis on the most critical content for the grade level. The National Council of Teachers of Mathematics (2006) released a set of focal points for each grade level designed to focus instruction on critical concepts for students to master within a specific grade. Similarly, the National Mathematics Advisory Panel (2008) detailed a route to preparing all students to be successful in algebra. In the lower elementary grades, the core focus of instruction is on building student understanding of whole numbers. As students establish an understanding of whole numbers, rational numbers become the focus of instruction in the upper elementary grades. Accordingly, screening measures used in the lower and upper elementary grades should have items designed to assess students understanding of whole and rational number conceptsas well as computational proficiency.
We stress that no one screening measure is perfect and that schools need to monitor the progress of students who score slightly above or slightly below any screening cutoff score.
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