Highlights for Math Intervention
Taken from DMA Research and Intervention Recommendations
By the end of Pre-School:
 Students will be able to count to 30 and count back from 5

Students will be able to tag 1-1 to 9
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Students will have cardinality (know the total number of objects counted without having to recount the group- up to 9)
 Students will be able to identify symbols up to 9
 Students will be able to write numerals to 9

Students will be able to find one more to 9

Students will be able to find one less from 5
 Students will be able to subitize finger patterns to 5
 Students will be able to compare and order 1-10

Students will be able to find partners to 5 (parts of 2 are 1 and 1; parts of 4 are 1 and 3, 3 and 1, or 2 and 2, etc.)
By the end of Kindergarten:
 Students will be able to count to 100 and count back from 10

Students will be able to tag 1-1 to 20

Students will move into a counted collection
 Students will be able to identify symbols up to 20
 Students will be able to write numerals to 20

Students will be able to find one more to 20

Students will be able to find one less from 20

Students will be able to subitize dot patterns to 5
 Students will be able to compare and order to 20

Students will be able to find partners to 10 (the parts that add up to 7, etc.)
By the end of Grade 1:

Students will be able to count to 120 and count back from 20
 Students will be able to tag 1-1 to 100
 Students will move into a counted collection

Students will be able to identify symbols up to 120

Students will be able to write numerals to 120
 Students will be able to find one, two, ten more to 100
 Students will be able to find one, two, ten less to 100

Students will be able to visualize patterns to 10

Students will be able to compare and order to 100
 Students will be able to find partners to 20 (the parts that add up to 15, etc.)

By the end of Grade 2:

Students will be able to count to 1000 and count back from 100

Students will be able to count by groups to 1000
 Students will be able to identify symbols up to 1000

Students will be able to write numerals to 1000

Students will be able to find one, two, ten, hundred more to 1000

Students will be able to find one, two, ten, hundred less to 1000
 Students will be able to visualize patterns to 20
 Students will be able to compare and order to 1000

Students will be able to make multiple representations of the same number

Students will be able to find the missing part of a number when given the whole and the other part
 Students will have a solid understanding of place value of whole numbers to 1000
Benchmarks for Critical Foundations of Algebra- Taken from the National Math Panel’s Report entitled: “A
Road Map for Mathematics Achievement for All Students”
(Highlights for Math Intervention)
By the end of Grade 3:

Students should be proficient with addition and subtraction of whole numbers
 Students should be developing an understanding of computational estimation and rounding
By the end of Grade 4:
 Students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals
 Students should be developing an understanding of computational estimation and rounding
By the end of Grade 5:
 Students should be proficient with multiplication and division of whole numbers

Students should be proficient with comparing fractions and decimals and common percents and with the addition and subtraction of fractions and decimals
 Students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (ie. Trapezoids)

Students should be developing an understanding of computational estimation and rounding
By the end of Grade 6:
 Students should be proficient with multiplication and division of fractions and decimals
 Students should be proficient with all operations involving positive and negative integers


Students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface areas and volumes
 Students should be developing an understanding of computational estimation and rounding
By the end of Grade 7:

Students should be proficient with all operations involving positive and negative fractions

Students should be proficient with all operations involving percent, ratio, and rate and extend this work to proportionality

Students should be familiar with the relationship between similar triangles and the concept of the slope of a line
 Students should have an understanding of computational estimation and rounding

RtI and Math- Tier 2 & 3 Content Focus and Suggestions for Instruction during Intervention
Tier 2 and 3 Mathematics Intervention Programs should focus on:

Proficiency with whole and rational numbers
 Understanding the reasoning for underlying operations

Building fact fluency

Identifying the structure of word problems
Note: Tier 2 & 3 Interventions do not necessarily need to align with the core curricula. It is not essential to cover all of the math topics in Intervention because students are exposed to them as a part of the regular curriculum. Intervention does need to emphasize the foundational mathematical proficiencies listed below.
Essential Topics and Key Strategies for Grades K-5:
 Whole Numbers o Strategic counting o Number composition o Solving problems involving whole numbers o Understanding place value o Understanding the underlying meaning of addition and subtraction operations
Essential Topics and Key Strategies for Grades 4-8:

Rational numbers o Operations with fractions, decimals, ratios and percents o More complex operations with whole numbers (i.e. long division)
Word Problems:

Intervention students need explicit instruction on how to solve word problems
 Intervention students need assistance with understanding problem structure
 Intervention students need assistance with understanding problem types and their solutions o Problems types:
 Change (involving adding or subtracting a quantity; often have an element of time involved)
 Compare (involve comparing quantities of items in different sets)
 Division (involve dividing a collection into groups or an item into parts) o Solutions:
 Students should be able to analyze word problems
 Students should be able to categorize word problems by type then apply the appropriate solution

Note: It may be useful to help students visualize or represent the problem pictorially o Students also need to be able to:
 Distinguish irrelevant information

 Have quick retrieval with math facts

If not, this will cause issues with new math concepts and word problems (for example, they can’t work with fractions without rapid recall of whole numbers)
 Tier 2 & 3 Interventions should include AT LEAST 10 minutes per day of practice with math facts o Note: Some research suggests that strategic approaches are more effective for learning and practicing facts than rote memorization o For younger students: explicit modeling of counting on or counting up; presenting facts in number families o For Gr. 2-8: teaching students to derive facts by using the associative, commutative, and distributive properties o Some ways to practice math facts include: flashcards, games, contests
&competitions, computer-supported instruction o Be sure to include cumulative and frequent review to build automaticity
What is systematic instruction and what does it look like?
 Teachers introduce concepts gradually and logically o Students are then given many opportunities to apply concepts
 Teachers use explicitness- giving clear explanations of concepts and using step-by-step modeling to show how to solve problems and perform operations o Teachers should discuss the reasoning behind each step as it is demonstrated (aka Think-
alouds)
 Teachers use materials that have many examples of both easy and difficult problems
 Teachers give students in Tiers 2 & 3 more guided/scaffolded practice o Teachers and students begin by solving problems together o As students begin to master skills they carry out more and more of the problem-solving on their own o Students are moved to independent problem-solving when they demonstrate little need for problem-solving support and are likely to experience success on their own o During practice:
 Students should be encouraged to talk out loud to peers and to the teacher about choices of strategies, reasoning behind problem-solving steps and solutions
 Teachers then provide corrective feedback (if needed) and additional guided practice and specific information on what was done right and what errors need to be corrected
 The review should be cumulative in nature o Since students often struggle with the meaning of abstract math symbols and the relationships between symbols and concepts, they can compensate by creating their own visual representations as part of problem-solving
 Teachers can use concrete materials and visual representations such as number lines, arrays, strip diagrams, graphs, and sample drawings to make math concepts and relationships explicit


Number lines for younger students: to demonstrate counting strategies, to teach principles of addition and subtraction, and to show and compare magnitude

Number lines for older students: can use both open and double number lines to perform operations with rational numbers
 For students to get the most out of visual and concrete representations, they need systematic and consistent exposure to examples
 CONCRETE OBJECTS  VISUAL REPRESENTATIONS  ABSTRACT SYMBOLS
Motivation:
Students in Tiers 2 & 3 need extra support to stay motivated: o Praise student effort o Praise student engagement in lessons o Praise completion of tasks o Praise accuracy of work o Praise is most effective when it is connected to specific accomplishments and when it recognizes effort as well as concrete progress o Students should monitor their progress and set short term goals for themselves
 This allows for more tangible rewards to remain motivated