The development of basic number sense is critical for a child's future understanding of more difficult math concepts. "A lack of sufficient mathematical skill and understanding affects one's ability to make critically important educational, life, and career decisions." Sherman, Richardson, and Yard
Diagnostic Tools (to help determine if a student is struggling with this skill):
Use Curriculum-Based Measurements to determine what skills the student is lacking.
AIMSweb Early Numeracy Curriculum-Based Measurements (Oral Counting, Number Identification, Quantity Discrimination and Missing Numbers)
Intervention Central's Early Math Fluency CBM Generator
Research-Based Interventions to try:
Activities to Increase Pre-Number Skills These strategies are a series of activities for developing pre-number skills including: one-to-one correspondence skills, which are a necessary prerequisite for counting objects; classification skills, necessary for grouping; and seriation skills, which help children understand concepts of quantity. These are very beginning activities to develop number sense. These activities should be used in hierarchical progression until solid skills are developed in each area. They may be varied and grouped throughout an intervention period to keep interest up. (Picture Cards)
Kinesthetic Activities to Increase Counting Skills These strategies are a series of activities for developing counting skills, including one-to-one correspondence skills, which are a necessary prerequisite for counting objects. These are very beginning activities to develop counting skills. These activities may be varied and grouped throughout an intervention period to keep interest up.
The Number Game This intervention strategy targets the specific skill of number identification. While counting and one-to-one correspondence are also included in this strategy, these are prerequisite skills that could be taught through other number sense activities. (Game board 0-10) (Game board 11-20) (Spinner)
Are These the Same? The concepts of “more,” “less,” and “same” are basic relationships contributing to the overall concept of number. Children begin to develop these concepts as early as age 2. An entering kindergarten student can usually choose the set that is more if presented with two sets that are clearly different in number. The concepts of less and same are more difficult to grasp, which is likely due in part to the fact that children have fewer opportunities to use these words in everyday life.
Progress Monitoring Tools:
AIMSweb Early Numeracy Curriculum-Based Measurements (Oral Counting, Number Identification, Quantity Discrimination and Missing Numbers)
Intervention Central's Early Math Fluency CBM Generator
Develop quizzes in Galileo to determine growth over time
The development of accuracy and speed of recall with basic mathematics facts so that higher order mathematical skills are stronger.
Diagnostic Tools (to help determine if a student is struggling with this skill):
Use Curriculum-Based Measurements to determine the skill deficit
AIMSweb Math Computation Curriculum-Based Measurement (M-COMP)
Intervention Central's Addition, Subtraction, Multiplication, Division and Mixed Skills Curriculum-Based Measurements
easyCBM Math Progress Monitoring (Computation is one of the strands assessed)
Research-Based Interventions to try:
Reciprocal Peer Tutoring (Worksheet for Peer Tutoring) The purpose of this intervention is to improve math performance and behavior during math instruction by means of peer tutoring, group rewards, and self-management procedures. Students work in groups to monitor their academic progress acting as instructional partners for each other, setting team goals, and managing their own group reward contingencies. Reciprocal peer tutoring has been demonstrated to improve math performance as well as students’ perceptions of their own academic competence and self-control. The intervention takes approximately 30 minutes – 20 minutes for peer tutoring and 10 minutes for individual class drills and checking.
Multicomponent Intervention for Math Fluency Fluency and accuracy with math facts is an important prerequisite to higher level math problem solving skills. When students are fluent with basic facts they are more motivated to persevere on difficult tasks and are more likely to be more confident when learning new math skills. This is a comprehensive multicomponent strategy which incorporates several individual empirically validated strategies into one comprehensive intervention approach. This strategy has been adapted from the work of Rhymer, Dittmer, Skinner and Jackson (2000) and includes explicit timing, positive practice, overcorrection, and performance feedback which are embedded within a peer tutoring structure.
Explicit Timing- Explicit timing has been shown to increase how quickly and accurately problems are worked. Through explicit timing exercises, students will become more automatic in math facts and thus become more proficient in solving math problems.
Addition Fact Families The following is a series of addition fact strategies that students can be taught to use early on that will help with remembering addition facts. These are adapted from the work of Thornton and Toohey (1985). This approach to learning basic facts is considered superior to just a memorization approach. (Doubles Pictures)
Subtraction Strategies These math strategies focus on subtraction. They were developed to assist students who are experiencing difficulty remembering subtraction facts, particularly those facts with teen minuends. They are beneficial to students who put heavy reliance on counting. Students can benefit from of the following strategies only if they have ability: to recognize when two or three numbers have been said; to count on (from any number, 4 to 9); to count back (from any number, 4 to 12). Students have to make themselves master related addition facts before using subtraction strategies.
Multiplication Fact Families The following is a series of multiplication fact strategies students can be taught to use early on that will help with remembering multiplication facts. These are adapted from the work of Wood and Frank (2000). This approach to learning basic facts is considered superior to just a memorization approach. By using the strategy method, less rote memorization is required. (Doubles Pictures) (Nines Links)
Please Excuse My Dear Aunt Sally-This mnemonic strategy is designed to help students remember computational order. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. (Cue Cards) (Poster) (Problems)
IES Practice Guide- Developing Effective Fraction Instruction for Kindergarten Through 8th grade
Progress Monitoring Tools:
AIMSweb Math Computation Curriculum-Based Measurement (M-COMP)
Intervention Central's Addition, Subtraction, Multiplication, Division and Mixed Skills Curriculum-Based Measurements
easyCBM Math Progress Monitoring (Computation is one of the strands assessed)
Develop quizzes in Galileo to determine growth over time
Diagnostic Tools (to help determine if a student is struggling with this skill):
Informal assessment data and observations indicating the student is struggling with word problems will help narrow in on the specific concern.
Research-Based Interventions to try:
SIR RIGHT This mnemonic strategy is designed to improve students’ math problem-solving skills. It cues students through the steps to follow to solve word problems effectively. (SIR RIGHT Cue Cards) (SIR RIGHT Poster)
Solving Word Problems Using Structured Organizers This intervention gives elementary and middle students better understanding of reading, interpreting, and solving word problems in mathematics. The use of structure organizers is first modeled by the teacher. Then students use the organizers on their own, which are then phased out as students become more proficient.
STAR for Algebraic Word Problems This mnemonic strategy is designed to improve students’ algebraic word problem-solving skills. (Cue Cards) (Posters)
Progress Monitoring Tools:
Collect daily work samples and use them to determine if the student is making progress.
Develop quizzes in Galileo to determine growth over time
The National Council of Teachers of Mathematics (NCTM) found in the 2001 National Research Council Report that 5 crucial strands of mathematics exist.
1. Conceptual Understanding:
Comprehension of mathematical concepts, operations, and relations
2. Procedural Fluency:
Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
3. Strategic Competence:
Ability to formulate, represent, and solve mathematical problems
4. Adaptive Reasoning:
Capacity for logical thought, reflection, explanation, and justification
5. Productive Disposition:
Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
(NRC, 2001, p. 5)
Diagnostic Tools (to help determine if a student is struggling with this skill):
Math Reasoning Inventory- MRI is an online formative assessment tool designed to make teachers’ classroom instruction more effective. The MRI questions focus on number and operations and are based on content from the Common Core State Standards for Mathematics prior to sixth grade. They are questions that we expect…and hope…all middle school students to answer successfully. There are three assessments in MRI—Whole Numbers, Decimals, and Fractions. Each assessment has two sections—the Interview and the Written Computation sections. The Interview (10–12 questions), done face-to-face, focuses on core reasoning strategies and understandings. We ask students questions, give them time to think, and listen to them explain. The Written Computation section is completed independently by students.
Multidimensional Assessment: Guiding Response to Intervention in Mathematics by Karen Koellner, Melissa Colsman and Rachel Risley for Teaching Exceptional Children, 2011.
Research-Based Interventions to try:
Reciprocal Peer Tutoring (Worksheet for Peer Tutoring) The purpose of this intervention is to improve math performance and behavior during math instruction by means of peer tutoring, group rewards, and self-management procedures. Students work in groups to monitor their academic progress acting as instructional partners for each other, setting team goals, and managing their own group reward contingencies. Reciprocal peer tutoring has been demonstrated to improve math performance as well as students’ perceptions of their own academic competence and self-control. The intervention takes approximately 30 minutes – 20 minutes for peer tutoring and 10 minutes for individual class drills and checking.
Hands-On Equations This activity is for any student experiencing difficulty with algebraic concepts. (Hands-On Equations Mat)
Free Time The purpose of this intervention is to increase the accuracy and completion rates of mathematics class work with a group-oriented free-time contingency.
IES Practice Guide- Improving Mathematical Problem Solving in Grades 4-8
Progress Monitoring Tools:
AIMSweb Math Concepts and Applications Curriculum-Based Measurement (M-CAP)
easyCBM Math Progress Monitoring (Based on NCTM strands of mathematics)
Develop quizzes in Galileo to determine growth over time
Diagnostic Tools (to help determine if a student is struggling with this skill):
Math Reasoning Inventory- MRI is an online formative assessment tool designed to make teachers’ classroom instruction more effective. The MRI questions focus on number and operations and are based on content from the Common Core State Standards for Mathematics prior to sixth grade. They are questions that we expect…and hope…all middle school students to answer successfully. There are three assessments in MRI—Whole Numbers, Decimals, and Fractions. Each assessment has two sections—the Interview and the Written Computation sections. The Interview (10–12 questions), done face-to-face, focuses on core reasoning strategies and understandings. We ask students questions, give them time to think, and listen to them explain. The Written Computation section is completed independently by students.
Research-Based Interventions to try:
Reciprocal Peer Tutoring (Worksheet for Peer Tutoring) The purpose of this intervention is to improve math performance and behavior during math instruction by means of peer tutoring, group rewards, and self-management procedures. Students work in groups to monitor their academic progress acting as instructional partners for each other, setting team goals, and managing their own group reward contingencies. Reciprocal peer tutoring has been demonstrated to improve math performance as well as students’ perceptions of their own academic competence and self-control. The intervention takes approximately 30 minutes – 20 minutes for peer tutoring and 10 minutes for individual class drills and checking.
Hands-On Equations This activity is for any student experiencing difficulty with algebraic concepts. (Hands-On Equations Mat)
Free Time The purpose of this intervention is to increase the accuracy and completion rates of mathematics class work with a group-oriented free-time contingency.
Solving Word Problems Using Structured Organizers This intervention gives elementary and middle students better understanding of reading, interpreting, and solving word problems in mathematics. The use of structure organizers is first modeled by the teacher. Then students use the organizers on their own, which are then phased out as students become more proficient.
STAR for Algebraic Word Problems This mnemonic strategy is designed to improve students’ algebraic word problem-solving skills. (Cue Cards) (Posters)
IES Practice Guide- Improving Mathematical Problem Solving in Grades 4-8
Progress Monitoring Tools:
AIMSweb Math Computation Curriculum-Based Measurement (M-COMP)
AIMSweb Math Concepts and Applications Curriculum-Based Measurement (M-CAP)
Develop quizzes in Galileo to determine growth over time
Illuminations by the National Council for Teachers of Mathematics (NCTM)
Intervention Central's Academic Interventions
Doing What Works (Scroll down to Math and Science)
Mathwire.com- Mathwire.com is designed to provide activities and appropriate worksheets for teachers to use in their classrooms. All activities and worksheets support the constructivist approach to learning mathematics and the NCTM Standards.
IES Practice Guide- Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools
IES Practice Guide- Developing Effective Fraction Instruction for Kindergarten Through 8th grade
IES Practice Guide- Improving Mathematical Problem Solving in Grades 4-8
Mathlanding- Resources and Tools for Elementary Math Specialists and Teachers
Rules to Lower the Amount of Memorization in Math
New Zealand Maths Resource Finder (Recommended by CDE)
Effective Mathematics Instruction by Kathlyn Steedly, Ph.D., Kyrie Dragoo, M.Ed., Sousan Arafeh, Ph.D., & Stephen D. Luke, Ed.D. for National Dessemination Center for Children with Disabilities
MathVIDS Video Instructional Development Source- Instructional Strategies, Videos and Information to help better understand math learning problems
Annenberg Learner- Teaching Math: A Video Library, K-4
Add Gayle's blog/classes page