Intervention: The Students Furthest Behind Made the Greatest Gains

The Intervention Model

More practice doesn't fix
an idea that was never built.

When a student is behind, the gap is rarely the grade-level procedure. It is a concept underneath it — often from an earlier year. Three moves close it.

A student stuck on adding fractions in Grade 5 is often missing an idea from Grade 3. Practicing the Grade 5 procedure again doesn't reach it. Intervention finds that layer and builds it.

Step 1

Understand the Gap

Targeted assessments and student responses help teachers identify mathematical ideas that may be limiting progress. Teachers examine how students understand quantities, units, place value, and relationships to determine where instruction should focus.

Step 2

Develop the Mathematics

Targeted activities help students develop mathematical relationships through visual models, precise language, and symbols. Students learn to compose, decompose, partition, and iterate units, connecting these structures to procedures and grade-level mathematics.

Step 3

Practice and Apply

Purposeful practice, explicit instruction, and feedback help students develop accuracy, efficiency, and flexibility. Students apply mathematical relationships and procedures across different problems and contexts, strengthening their understanding and fluency.

The Research

What Does the Research Tell Us
About Students Who Struggle?

A peer-reviewed, cluster-randomized study involving 25 schools and 4,618 students found the largest gains among students who started lowest. In an independent evaluation by RMC Research, 79.2% of students in DMT classrooms reached proficient or advanced, compared with 65.7% in comparison classrooms. The DMT schools served higher proportions of students experiencing poverty and English learners.

79.2%

Proficient or Advanced, DMT classrooms
vs 65.7% comparison (RMC Research)

4,618

Students in the randomized study
25 schools, cluster-randomized

350,000+

Students reached
across 300+ schools

DMT schools were ahead in every grade, 2 through 5. These are the unadjusted averages from Table 2 of Brendefur et al. (2021). After adjusting for students' prior achievement, the overall effect was +0.09 SD.

Research and evidence: Brendefur et al. (2021), International Journal of Science and Mathematics Education — cluster-randomized, 25 schools, 4,618 students; doi.org/10.1007/s10763-021-10184-z. Independent evaluation by RMC Research: 79.2% vs 65.7% proficient or advanced. School results: Iowa Elementary, Nampa, Idaho (Grade 5, Idaho SBAC); West Canyon Elementary (English learners, Idaho ISAT); Whittier Elementary (school-reported honors participation).

The Mathematics Behind the Approach

See the mathematics students build.

Math Success begins with quantities and units students can see. Here is 3 × 4 — not a fact to memorize, but three iterations of a unit of 4, on a proportional bar model and a number line. Every operation students learn connects back to this structure.

Ask: What is the unit being repeated, and how does the model show the total quantity? This same reasoning — units, composition, iteration — carries from early counting through fractions and algebra.

The same three iterations of a unit, whether the unit is 4, one fourth, or x. Students who build the structure with whole numbers recognize it in fractions and algebra.

The DMT Framework

Five Practices That Strengthen
Mathematics Teaching and Learning.

The DMT framework builds on more than 40 years of research in mathematics teaching and learning. Developed through the work of Jonathan Brendefur, PhD, and DMTI researchers and educators, the framework connects research with classroom instruction.

01 Taking Students' Ideas Seriously Teachers listen carefully to students' ideas and use their responses to understand how they are reasoning about mathematics. These ideas help teachers select questions, models, and tasks that extend student thinking.

02 Using Multiple Strategies and Models Students use and connect strategies and models to make sense of mathematical relationships. Concrete materials, mats, bar models, number lines, and area models support the development of ideas across grades, with representations selected to match the mathematics and students' understanding.

03 Developing Conceptual Understanding and Procedural Fluency Students develop understanding and fluency together as they connect mathematical relationships, visual models, language, and procedures. Explicit instruction, feedback, and purposeful practice help students become accurate, efficient, and flexible.

04 Using the Structure of Mathematics Students learn to recognize quantities and units and use composition, decomposition, partitioning, and iteration to understand mathematical relationships. These structures connect ideas across number, operations, fractions, measurement, geometry, and algebra.

05 Embracing Misconceptions and Mistakes Mistakes and unexpected responses provide opportunities to investigate student thinking. Teachers use these responses to ask questions, examine mathematical relationships, and help students refine their understanding.

Three Pillars

What the framework stands on.

The Mathematics That Matters Most

Spatial reasoning, measurement, and number are deeply connected. Students develop mathematical understanding as they reason about quantities, units, and relationships. Visual and linear models help students see these structures and connect them to mathematical symbols and procedures.

The Psychology of How Students Learn

Students develop and refine mathematical ideas as they connect experiences, models, language, symbols, and procedures. Instruction supports students in progressively formalizing these ideas while providing opportunities for explanation, feedback, and practice.

Instruction That Responds to Student Thinking

Teachers investigate student thinking, select purposeful mathematical tasks, and use questions, models, and feedback to extend understanding. Student responses inform instructional decisions and help teachers determine when to revisit, extend, or formalize mathematical ideas.

1 · Experience

Build 7 + 5 as a bar. Three of the five complete the ten, and two are left over.

2 · Model

Show the same move on a number line: up 3 to 10, then 2 more.

3 · Language

Say what the model shows, in precise language.

4 · Symbols

Write the relationship the model and the words describe.

5 · Procedure

Recall it fluently, and use the same move with larger numbers.

The psychology pillar on one problem. 7 + 5 moves from experience to model, language, symbols and procedure, and each step connects to the one before.

What Your School Receives

Assessments, activities and models, and support for your teachers.

Targeted Assessments

Assessments that investigate how students understand quantities, units, place value, and relationships — evidence teachers use to target instruction to the mathematics underneath the gap.

Instructional Activities and Models

Classroom activities and mathematical models that develop the ideas underneath the gap and connect them to grade-level mathematics — with practice that builds accuracy, efficiency, and flexibility.

Professional Learning and Coaching

Professional learning and in-classroom coaching that help teachers put the framework to work with their own students, from modeling lessons to ongoing instructional feedback.

One assessment item, and what the answer shows. The student's mistake points to the idea to build, and the bar model builds it.

How We Work With Teachers

Supporting Teachers
Where Mathematics Learning Happens.

We work alongside teachers to examine student thinking, strengthen mathematical understanding, and refine instructional practices. Through modeling, co-teaching, and feedback, teachers develop ways to use mathematical models, ask purposeful questions, and connect student understanding to procedures and practice.

1

Start from where teachers are

No prerequisites. We build from what your teachers already understand and value about their own students.

2

Discover the high-leverage moves

Together we find the small instructional decisions — the question asked, the model chosen, the language named — with the biggest effect on student thinking.

3

Stay with them all year

Ongoing support may include classroom modeling, co-teaching, instructional coaching, professional learning sessions, and access to DMTI resources, depending on the school's partnership.

Schedule a Free Demo

A DMTI math specialist co-teaching with a classroom teacher

Modeling, co-teaching, and coaching in the classroom — not a workshop teachers forget.

Real Results

Where it was tried, it moved.

Principals and instructional leaders from their own schools. More results →

+39 pts

Grade 5 Mathematics Proficiency · Iowa Elementary, Nampa, Idaho

From 16% to 55% proficient over four years — a 39-percentage-point increase.

“The co-taught professional development allowed our building to make tremendous improvements in mathematical instruction.”
— Chance Whitmore, Principal

5% → 30%

Mathematics Proficiency · English Learners · West Canyon Elementary

From 5% to 30% on the Idaho state assessment (ISAT) over four years.

“Implementing the DMT lessons and framework has influenced a lot of change for us. The problem solving that the program requires of students really challenges them to think critically about the math.”
— Cindy Dodd, Principal

“

Mountain Home School District (Idaho)

A district-wide DMTI partnership.

“Working with the DMTI professionals has been a game changer for our district… they have helped our teachers become better all-around teachers.”
— Jeff Johnson, Director of Instruction

0 → 30

Students in 7th-Grade Honors · Whittier Elementary

School leaders reported an increase from zero to approximately 30 students participating in seventh-grade honors mathematics following implementation of DMTI instructional practices — at a Title I school in a refugee community.

From real classrooms.

“My entire 1st grade class was at grade level by the end of the year!”

Kelli Rich

1st Grade Teacher

“I wanted to share my math ISAT percentage with you. 73%! That's over 20 points higher than last year!”

Mrs. Johnson

Elementary Teacher

Next Step

See How Math Success Can
Strengthen Intervention at Your School.

In a free 15-minute demonstration, see how our assessments, instructional activities, mathematical models, and teacher support work together to develop student understanding and fluency.