When considering instructional activities for mathematics, teachers should focus on strategies that have evidence to support their effectiveness. The Virginia Department of Education, building on the work of the Institute for Education Sciences (IES), has identified five evidence-based strategies to specially-design mathematics instruction: 1. **Explicit Instruction **2. **Formal Mathematical Language** 3.** Concrete, Representational, and Abstract Connections** 4. **Fact and Computational Fluency **5. **Problem Solving**.

## Explicit Instruction

**What Is It?**

Explict Instruction has three main components: ** Modeling**,

**, and**

*Practice***(VDOE, 2020).**

*Supports***Modeling: **Facilitated by the teacher, this component prepares students to complete mathematics skills successsfully by providing clear explanations and planned examples.

**Practice:** Involving both the student and the teacher, practice provides students with multiple opportunities to apply knowledge of a learned concept. This includes both guided and independent practice.

**Supports:** Teachers provide support to students during both guided and independent practice by asking high- and low-level questions, providing students with multiple opportunities to respond, providing immediate affirmative and corrective feedback, and maintaining a brisk pace.

*Where’s the Evidence?*

The What Works Clearinghouse (2021) has assigned explicit and systematic instruction with a rating of ** strong level** of evidence of effectiveness based on 43 case studies.

## Formal Mathematical Language

**What Is It?**

Formal mathematical language refers to the precise mathematical terms used to describe concepts and procedures (VDOE, 2020). To promote students’ understanding of mathematical language, teachers should: 1. Use formal mathematics vocabulary terms 2. Use similar or related terms correctly and precisely 3. Plan for language use prior to instruction 4. Include explicit vocabulary activities in instruction 5. Hold students accountable for using formal mathematical language correctly.

*Where’s the Evidence?*

The What Works Clearinghouse (2021) has assigned formal mathematical language with a rating of ** strong level** of evidence of effectiveness based on 16 case studies.

## Concrete, Representational, and Abstract Connections

**What Is It?**

The Concrete-Representational-Abstract (C-R-A) sequence is a framework for teaching mathematics concepts that is utilized to facilitate students’ deeper understanding of mathematics concepts (VDOE, 2020).

The **Concrete **stage of the sequence refers to the use of physical, three-dimensional materials and objects. This includes manipulatives such as counters, base-10 blocks, fraction bars, and algebra tiles.

The **Representational **stage of the sequence includes two-dimensional drawings, images, and virtual manipulatives. This stage is also referred to as pictorial or semi-concrete.

The **Abstract** stage of the sequence consists of numbers, symbols, and words. This stage is typically the “destination”, while the concrete and representational stages support students’ understanding to arrive at the destination.

*Where’s the Evidence?*

The What Works Clearinghouse (2021) has assigned the use of concrete and semi-concrete representations with a rating of ** strong level** of evidence of effectiveness based on 28 case studies.

## Fact and Computational Fluency

**What Is It?**

Computational fluency is the ability to think flexibly in order to choose appropriate strategies to solve problems accurately and efficiently (VDOE, 2020).

Strategies to build computational fluency include: 1. Using explicit instruction to model and practice different computational algorithms 2. Providing students with opportunities for spaced practice 3. Engaging students in fluency practice. This includes brief, daily opportunities for fact practice. *Facts should be practiced in a variety of ways, including numeracy routines, games, activities, and songs.*

*Where’s the Evidence?*

The What Works Clearinghouse (2021) has assigned the use of timed activities as one way to build students’ fluency with a rating of ** strong level** of evidence of effectiveness based on 27 case studies.

## Problem Solving

**What Is It?**

Teaching strategies for solving word problems is essential for students with mathematics difficulties. In order to create and solve problems from real-world data, students need to develop a set of skills and strategies for solving a range of problems (VDOE, 2020).

Strategies for teaching problem solving include: 1. Teaching students an attack strategy to guide the process of problem solving 2. Teaching students to recognize and solve word problems according to the schema of the problem 3. Utilizing appropriate mathematical language to help students understand the meaning of each word in a problem.

**It is important that teachers DO NOT tie key words to operations in word problems because not only is it an ineffective practice, it can lead to incorrect responses and diminishes conceptual understanding.**

*Where’s the Evidence?*

The What Works Clearinghouse (2021) has assigned deliberate instruction on word problems to deepen students’ mathematical understanding with a rating of ** strong level** of evidence of effectiveness based on 18 case studies.

### Resources

Below are resources that can be utilized to dive deeper into the five evidence-based strategies for mathematics instruction.

VDOE’s Evidence-Based Specially Designed Instruction in Mathematics Resource Guide

Assisting Students Struggling in Mathematics: Intervention in the Elementary Grades (IES/WWC)

VDOE’s Students with Disabilities in Mathematics Frequently Asked Questions

IRIS Center: Evidence-Based Mathematics Practices

#### References

Fuchs, L.S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K.S., Woodward,

J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., and Morgan, S. (2021). Assisting

Students Struggling with Mathematics: Intervention in the Elementary Grades (WWC 2021006).

Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of

Education Sciences, U.S. Department of Education. Retrieved from http://whatworks.ed.gov/.

Virginia Department of Education (2020). Evidence-Based Specially Designed Instruction in Mathematics Resource Guide. Retrieved from https://www.doe.virginia.gov/special_ed/disabilities/learning_disability/swd-mathematics-resources.pdf