# Evidence-based math instruction: What you need to know

You’ve probably heard one or more students say “I’m not a math person.” Some may have had experiences that made them dislike math. Others have the idea that they can’t be good at math because of their gender.

But some students who say they’re not “math people” might be among the 25–35 percent of students who truly struggle with math. They could even be among the 5–8 percent of students who have significant challenges with math.

Yet, we know that a strong foundation in math is important for all students. It opens doors to higher-level math courses and to careers in the STEM field.

The good news is that there are effective, evidence-based ways to teach math that can be used with your current curriculum. You can use these teaching practices to help every student learn math at all grade levels — from addition to algorithms.

## What is evidence-based math instruction?

There are four elements that make up effective math teaching.

#### 1. Explicit instruction with cumulative practice

**What it is:*** *Explicit instruction is a way of teaching that makes the learning process completely clear for students. With explicit instruction, you model a skill and verbalize your thinking process, using clear and concise language. You give students opportunities for guided and independent practice — including practicing the new skill and reviewing skills that they’ve learned in the past. You also give students quick feedback so they stay on track.

Research has shown that using explicit math instruction can improve students’ ability to perform operations and solve word problems.

**Why it helps: **When you use this practice, you model the skill so clearly that students don’t need to guess what they have to do. The cumulative practice of explicit instruction is especially helpful because it keeps old skills fresh in students’ minds. That’s a big plus for students who struggle with working memory. Repeated practice of related skills done over time helps them to quickly retrieve information and keeps up their math fact fluency.

Making sense of instructions (such as multi-step math problems) is a challenge for students who have trouble with language processing. Math requires students to process a lot of language, both oral and written. Explicit instruction can reduce how much of that processing a student needs to do.

**Teaching tips**

Share a clear learning goal with students.

Use “do nows” that activate prior knowledge, such as a beginning-of-the-class word problem to revisit a skill you taught the day before.

Give a crystal clear explanation of the skill or strategy.

Use multiple examples to show all the steps. (In some cases, use different examples, like some addition problems with the greater addend first and other addition problems with the lesser addend first.)

Do a think-aloud to verbalize your thinking.

Allow lots of opportunities for guided and independent practice.

Vary how you ask for student responses, such as verbal (like choral response), written (like stop-and-jot), and nonverbal cues (thumbs up/thumbs down).

Include previously learned skills in practice opportunities.

Give students immediate feedback.

#### 2. Visual representation

**What it is:** Visual representation is a way for students to *see *math. You can visually represent math using number lines, tape diagrams (also known as bar models), pictures, graphs, and graphic organizers.

Visual representation is often used in an instructional approach called concrete-representational-abstract, or CRA. (Some teachers call this approach concrete-pictorial-abstract.)

With CRA, you use visual representations to help students understand abstract math concepts. For example, students can use concrete like Unifix cubes to solve an addition problem. (Even though concrete manipulatives are more commonly used in elementary classrooms, they can help older students, too.)

Students could also use drawings or pictures (again, more visuals) to show the same math concept. For example, students might use tally marks to represent the addition problem. At the same time, you could show the same concept in the abstract as an addition sentence with numbers and symbols.

With CRA, you don't need to follow the order of concrete, then representational, then abstract. You might have students use concrete tools while you model the same concept in the abstract. Also, you can go back and forth between the three representations based on your students’ needs.

**Why it helps: **Visual representations help all students understand abstract math concepts and solve problems. These representations can remove language barriers related to word problems for students who learn and think differently, as well as for English language learners.** **

When students create their own visual representations, they have a way to show their understanding (or misunderstanding that you can then correct). In fact, research shows that students who use accurate visual representations are six times more likely to correctly solve problems compared to students who do not use them.

Note:This strategy may not be as useful for students who struggle with math because of difficulties with spatial reasoning or visualization. It can also pose challenges for students who are blind or who have low vision. For these students, you can make accommodations such as using large-print texts or tactile graphics.

Tactile graphics are simple 3D representations of spatial ideas, such as pictures, maps, graphs, diagrams, and other images. They use raised lines and other tactile features to provide the information your student needs to learn. Talk with your district’s consultant for the visually impaired about how to teach students to use these tools.

**Teaching tips**

Teach students to use number lines, tape diagrams, pictures, graphs, and math graphic organizers.

Encourage students to use visual representations to show their mathematical thinking, like using a hundreds chart while they practice skip counting.

Introduce concepts and skills using concrete manipulatives, like using base 10 blocks to teach place value.

Show concepts and skills using representations and pictures, like tallies, dots, and circles.

Model concepts and skills at the abstract level, like using numbers and symbols.

Provide students with practice opportunities at each stage.

#### 3. Schema-based instruction

**What it is: **One of the most effective ways to help students solve word problems is to teach them to identify word problem types. According to research, this practice is especially helpful for students who struggle with math.

You can explicitly teach students to recognize the patterns in word problems. This is called schema-based instruction, meaning that students use what they know about patterns in word problems to solve the problem. There are two main types of schemas: additive and multiplicative. Additive includes addition and subtraction problems. Multiplicative includes multiplication and division problems.

Consider this word problem: “Erin has 4 crayons. Aleysha has 3 crayons. How many crayons do they have together?*”* Students might recognize that it follows a pattern of having two sets of items and the word “together.” That can lead students to recognize an additive schema.

After students identify the schema, they can represent the information using a diagram or an equation before solving.

**Why it helps: **Students who struggle in math can have difficulty recognizing patterns and relationships in new situations. (This is known as fluid reasoning.) Schema instruction explicitly teaches students how to identify patterns as a way to connect it to the correct strategy to solve that type of word problem.

Students who learn and think differently, especially those who have challenges with executive functioning skills__,__ may also have difficulty with working memory and multi-step directions. Research shows that students who were taught using schema-based instruction were better able to solve both familiar and new multi-step problems.

**Teaching tips**

Teach students to analyze a word problem and

__identify the pattern__.Identify for students the unique features of each type of problem.

Explicitly teach the math vocabulary needed for that problem.

Show how to represent the information using a concrete representation first and then a visual representation.

Show multiple ways to solve the same problem.

#### 4. Peer interaction

**What it is:*** *With peer interaction, you pair up students to work together and have discussions about math. For example, students might do independent practice and then meet up with a partner to share what they learned. You can use flexible grouping to match up students, like pairing students with similar math abilities or by different strengths.

**Why it helps:*** *Peer discussions can develop students’ math language and vocabulary, as well as helping students express their reasoning. It can also help them become more aware of problem-solving processes — both how they solved the problem and how others solved it. Students who struggle with math may find this routine helpful because their peers may be able to explain a concept in a way they better understand. All students can benefit from seeing that the same problem can be solved in multiple ways.

**Teaching tips**

Pre-teach how to have peer-to-peer discussions. Work as a class to establish rules (like not talking over one another) and discussion prompts (like “What can you add to my explanation?”).

Encourage students to compare the ways they solved a problem and discuss the differences in their approaches.

Build in time at the end of your lessons for students to reflect with each other about their independent practice opportunities. You can even just put up a piece of student work and ask students, “What do you notice?”

Allow time for students to do math work with partners or in small groups.

## Who does evidence-based math instruction help?

Effective math teaching can help all students — but it’s particularly useful for students who struggle with math.

Students with __dyscalculia____,__ a learning disability that affects math, may have difficulty understanding number-related concepts. For instance, they may struggle to understand that the numeral 5 is the same as a group of five items and as the word *five*. They might also have trouble using math symbols or understanding math-related concepts like *greater than* or *less than*.

Evidence-based math instruction helps students with dyscalculia because it provides them with the explicit guidance and scaffolding they need to help them gain skills and understanding.

Students may also struggle with math because of trouble with executive functioning. Math requires students to pay attention to details, plan, and self-monitor. Students also have to keep track of steps — and maybe even change direction while they work. Evidence-based math instruction helps these students because it breaks problems into multiple steps and reduces distractions.

## How do I get started with evidence-based math instruction in my classroom?

Your school’s math curriculum may not include all of these elements. Or maybe your training didn’t cover this type of math teaching. If you want to have a better understanding of these strategies, you can also advocate for professional development in your school on this topic.

But even without formal training, you can try out any one of the elements of evidence-based math instruction in your classroom. For example, if a student is having trouble with the concept of place value, begin by providing base 10 blocks to show how two hundreds blocks, three tens blocks, and four ones blocks make the number 234 (concrete). Then move on to having the student draw it (representational), and finally move to 200 + 30 + 4 = 234 (abstract).

Explore these strategies that include key elements of evidence-based math instruction:

## How can families support this at home?

Common Core math standards. “New math.” These are just some of the terms families may have heard about math. Before your parent-teacher conferences, share this checklist (Spanish version here) with families.

It was created by Understood Teacher Fellow Pauli Evanson. The checklist will help families prepare their questions about math ahead of your meeting.

Once you start using evidence-based math instruction, you’ll begin to see that best practices that support students who struggle with math are also good for *all *students.