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The literature findings indicate:

Children who have had less experience or exposure to mathematical concepts and numeracy are at risk for mathematics failure.

Math is highly proceduralized and continually builds on previous knowledge for successful learning. Hence, early deficits ahve enduring and devastating effects on later learning.

Early mathematics interventions can repair deficits and prevent future deficits.

Instructional practices:

Explicit systematic instruction within authentic contexts. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, correct feedback, and frequent cumulative review.

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).

The "Big Ideas" enable teachers to build a true understanding of the strands in their students. This helps maintain and build connections for the way students think about mathematics, because the "Big Ideas" cut across every lesson and are not lesson specific.

Mathematics instruction needs to address the 4 essential domains (problem-solving; arithmetic skill/fluency; conceptual knowledge/number sense; reasoning ability).

Domain

Description

Guiding Questions

Problem-solving

Teach students to use knowledge of arithmetic, concepts and reasoning to solve math problems. Math's utility is in its promise to make work easier, build approaches to real need and in expanding the mind to see new solutions.

Most students are not asked to discuss their thinking about their approach to math solutions. How can you re-construct your style so that students imitate or initiate this transparent discussion about the meta-steps to problem solving?

How are you making parents aware of ways to approach math problem solving? Are you providing examples for parents to practice with students at home?

Do your teachers have the depth of knowledge necessary to pose mathematical questions and problems that are productive for students' learning?

Most teacher questions are posed with less than a three-second delay between question and expected student response. How might you invite prolonged thinking time for students to solve a problem in class.

Arithmetic skill/fluency

Teach students efficient and accurate methods for computing and using computational algorithms and math facts, and provide the necessary practice to build automaticity. A solid grounding in arithmetic is a prerequisite to learning advanced mathematics. Procedural fluency is necessary for understanding concepts and solving problems accurately and efficiently.

How explicit are you abut giving your students these essential math rules in order to help them recognize and trust the ways numbers behave?

What are the multiple ways that elementary, middle, and high school math teachers teach the rules of arithmetic?

Do you provide distributed practice for procedures and facts?

Do your parents know these rules and table so that they can support your students' study of the math language?

How are these rules incorporated into a coherent and repeated series of expectations for which students take responsibility?

Conceptual Knowledge/number sense

Teach students the ways that numbers behave and help students build their understanding of math concepts and vocabulary. Math is often a subject of disconnected content and random study. Students need to understand why math operations function the way they do and be able to explain the patterns they find.

How are math concepts and operations taught? Are they made explicit to students?

How are your students taught to understand number relationships?

Does your math instruction emphasize symbolic notation, diagrams and procedures?

Do you explicitly teach math vocabulary in conjunction with concept development?

Do your teacher have the depth of knowledge necessary to provide accurate explanations that are comprehensible and useful for students?

How do you clear about which concepts are expected to be know at which grades?

Reasoning Ability

Teach mathematical reasoning skills. Math is about using logic to explain and justify a solution to a problem. It is the mental muscle necessary to successfully explore puzzles. It can also extend something known to something not yet known.

How do your questions prompt students to explain and justify logical solutions to problems.

How do you address faulty reasoning by students?

Do your teachers have the depth of knowledge necessary to respond productively to students' mathematical questions and curiosities?

What opportunities do students have to provide logical verbal and written explanations of their reasoning?

How often are students asked to apply spatial reasoning to patterns and comparisons?

Assessing Math Performance

Select the assessment material: Is there sufficient information to determine what the student knows and is able to do?

Sample the student's performance: What skills did the student orchestrate given the different math dimensions?

Math instruction: What areas of math are in need of immediate support? What prerequisite skills are needed?

Teach the student: What fine-tuning needs to occur to ensure ongoing student success?

Determining prerequisite skills

Do a task analysis and determine what prerequisite skills is the student missing.

Teach the prerequisite skills and help the student make the connection to what they need to be learning.

There are times, you will just need to do an error analysis. The student has the skills to learn the concept, but he or she just keeps making the same mistake over and over.

Error Analysis

Score the problems

Begin with the first incorrect problem and attempt to determine how the student is obtaining the answers.

Use the pattern and see if the second incorrect problem follows the pattern and if it does not revise the your prediction.

Confirm the pattern using the other incorrect problems.

describe the student's strength and the error pattern.

Design the intervention plan.

Intervention Recommendations from What Works Clearinghouse

Recommendation

Suggestion

For students in kindergarten through grade 5, interventions should focus on in-depth coverage of properties of whole numbers and operations. Some older students may also need this in-depth coverage.

Intervention materials would typically include significant attention to counting, number composition, and number decomposition (to understand place-value multi-digit operations). Interventions should cover the meaning of addition and subtraction and the reasoning that underlies algorithms for addition and subtraction of whole numbers, as well as solving problems involving whole numbers. This focus should include understanding of the base-10 system (place value).

For students in grades 4 through 8, interventions should focus on in-depth coverage of rational numbers as well as advanced topics in whole number arithmetic.

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), and solving problems with fractions, decimals, ratios, and percents.

Interventions should include instruction on solving word problems that is based on common underlying structures.

Teach students about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solutions for each problem type.

Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems

Intervention materials should include opportunities for students to work with visual representations of mathematical ideas.

Use visual representations such as number lines, arrays, and strip diagrams.

If visuals are not sufficient for developing accurate abstract thought and answers, use concrete manipulatives first. Although this can also be done students in upper elementary and middle school grades, use of manipulatives with older students should be expeditious because the goal is to move toward understanding of --- and facility with --- visual representations, and finally, to the abstract.

Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.

For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts.

Teach students in grades 2 through 8 how to use their knowledge of properties, such as commutative, associative, and distributive law, to derive facts in their heads.

Include motivational strategies

Because many of the students struggling with mathematics have experiences failure and frustration by the time they receive an intervention, use tools that can encourage active engagement of students and acknowledge student accomplishments.

Reinforce or praise students for their effort and for attending to and being engaged in the lesson.

Consider rewarding student accomplishments.

Allow students to chart their progress and to set goals for improvement.

Math Resources Some resources (not a limited list) of things that you can use to provide instructional interventions include:

Resource

Areas

Description

Publisher/
Company

Cost

Number & Operations

Measurement

Geometry

Data Analysis & Probability

Algebra

Other

*Intervention Central

X

X

X

X

www.interventioncentral.org (click on academic resources) Intervention Central provides a variety of researched-based intervention resources.
Directions:

Alternative Math Techniques
Written by: Richard Cooper

X

X

X

X

X

X

Collection of alternative techniques for teaching arithmetic operations and math concepts.

Sopris West

$$

*One-Minute Academic Functional Assessment and Interventions
Author: Joe Witt & Ray Beck

X

Provides a functional assessment procedure designed to help teachers determine if the student CAN’T DO IT…OR WON’T DO IT. Then helps to link assessment data with interventions

Sopris West

$$

*RtI Intervention Manual

X

A compilation of collected interventions that are represented through various levels of intensity and categories

Envision Math Diagnosis & Intervention System (K-5)

X

X

X

X

X

Provides a diagnosis test and specific intervention lesson for intervention on identified skills.

Math Adoption

Free

Destination Math

X

Through sequenced, prescriptive step-by-step instruction, Destination Math helps students develop fluency in critical skills, math reasoning, conceptual understanding and problem-solving skills

RiverDeep

$$

Targeted Mathematics Intervention

X

A leveled intervention program that includes easy-to-follow lessons, pacing plans, vocabulary activities, diagnostic tests, and strategies. Each level is purchased separately.

The literature findings indicate:

Instructional practices:

The "Big Ideas" enable teachers to build a true understanding of the strands in their students. This helps maintain and build connections for the way students think about mathematics, because the "Big Ideas" cut across every lesson and are not lesson specific.

## Mathematics instruction needs to address the 4 essential domains (problem-solving; arithmetic skill/fluency; conceptual knowledge/number sense; reasoning ability).

DomainDescriptionGuiding QuestionsAssessing Math PerformanceDetermining prerequisite skills

There are times, you will just need to do an error analysis. The student has the skills to learn the concept, but he or she just keeps making the same mistake over and over.

Error Analysis## Intervention Recommendations from

What Works ClearinghouseRecommendationSuggestionWhat Works ClearinghouseTo learn more, review the following documents:TopicDocumentMath ResourcesSome resources (not a limited list) of things that you can use to provide instructional interventions include:Company

Directions:

Score Sheet:

Written by: Richard Cooper

Author: Joe Witt & Ray Beck

File