What evidence-based mathematics instructional practices can educators implement?
Page 5: Explicit Instruction
Explicit instruction involves teaching a specific concept or procedure in a highly structured and systematic manner. Research has indicated that teaching mathematics in this manner is highly effective and can significantly improve a student’s ability to perform mathematical tasks (e.g., adding, identifying equivalent fractions, finding the square root) and solve word problems. All students benefit from explicit instruction; however, students with MLD, ELs, and those from economically disadvantaged backgrounds often require it to learn foundational grade-level skills and concepts.
Research Shows
- An influential meta-analysis of mathematics interventions indicated that explicit instruction led to large improvements in student mathematics skills.
(Gersten et al., 2009) - The inclusion of explicit instruction in core mathematics instruction for kindergarteners improved achievement.
(Doabler et al., 2015) - Research across more than five decades has shown that explicit instruction is effective for different academic areas, including mathematics.
(Hughes et al., 2017)
This model is often referred to as explicit, systematic instruction. The term explicit refers to how the lesson is implemented, while the term systematic refers to the planning of well-structured and sequenced lessons. There are 16 components of explicit instruction that educators must consider throughout the different phases of developing and implementing lessons. Explore the components in each of the phases below.
To ensure they are prepared to teach engaging lessons and provide support for all students, educators should spend time systematically planning lessons. They can accomplish this by implementing the following explicit instruction components.
| Explicit Instruction Component | Description |
|
Identify critical instructional content |
This can consist of factual information, academic language, skills, strategies, and rules. |
|
Sequence skills logically |
Educators should plan lessons so that they teach skills that build on each other and move from:
|
|
Break complex skills and strategies into smaller instructional units |
This is also known as task analysis. Note in the following task analysis for adding two two-digit numbers that the skill is broken into four steps. Step 1: Add the numbers in the ones column. |
|
Design organized and focused lessons |
To ensure that lessons are efficient and effective, educators should address five questions:
|
|
Help students organize knowledge |
To help students understand how concepts and skills are connected, educators should:
|
Although this phase is tempting to skip, educators should make sure to take two to three minutes of instructional time to introduce the lesson. This sets the stage for student learning because it:
- Sets the tone for the lesson
- Identifies critical content
- States how information will be presented, evaluated, and practiced
- Provides expectations for student engagement
| Explicit Instruction Component | Description |
|
Provide a clear statement of goals and expectations |
Prior to each lesson, educators should take a few minutes to:
x
rationale for learning glossary x
learning goal glossary x
advance organizer glossary x
expectations glossary |
|
Review background knowledge |
Educators should assess students’ prior skills and knowledge to ensure they have the prerequisite skills needed to fully engage with and learn the content being presented (e.g., general knowledge of the topic, specific vocabulary terms, and skills or strategies). |
Educators must present the content—information, vocabulary, skills, strategies, and rules—in a way that students can understand and master them. They can do this using the explicit instruction components in the table.
| Explicit Instruction Component | Description |
| Use modeling
x
modeling glossary |
Educators demonstrate while explaining the critical content being taught. Demonstrating helps students understand how to perform each step. Using think-alouds to explain helps students understand why each step is important.
x
think-aloud glossary |
|
Use clear and concise language |
To help prevent confusion and increase retention, educators should use simple, unambiguous wording when presenting content. |
|
Provide examples and non-examples |
Providing a wide range of examples and non-examples can help students gain a deeper understanding of a new concept. Examples help students understand the attributes of the concept, while non-examples help students define why something does not apply to the concept. |
|
Maintain a brisk pace |
By maintaining a brisk pace, educators can optimize instructional time, the amount of content presented, and student engagement. |
After presenting the content, students need to practice applying the new knowledge. Initially, they need supports until they have a stronger grasp of the content, then those supports can be gradually removed (i.e., faded). To provide support, educators can implement the next two components of explicit instruction. Guided practice occurs after the teacher presents the content, while opportunities to respond occur during modeling and guided practice.
| Explicit Instruction Component | Description |
|
Offer guided practice x
guided practice glossary |
This involves students practicing previously modeled or taught skills while the educator offers support and feedback. Educators can provide support using prompts, or scaffolds, which can be physical (e.g., manipulatives), verbal (e.g., cues), or visual (e.g., graphic organizer). |
|
Provide frequent opportunities to respond x
opportunities to respond glossary |
When using this strategy, educators question students and they respond. This increases student engagement and allows educators to check for understanding. Questions can be directed at an individual student or a group, and student responses can be verbal or non-verbal (e.g., students hold up whiteboards with answers written on them). |
The first two of these components, monitor performance frequently and provide positive and constructive feedback, take place throughout instruction. On the other hand, distributed practice and cumulative practice occur after the lesson has been taught; however, teachers need to plan for this component in advance to ensure that it happens.
distributed practice
glossary
cumulative practice
glossary
| Explicit Instruction Component | Description |
|
Monitor performance frequently |
Educators should carefully observe students’ responses and engagement during the entire lesson. This allows them to gauge learning for the whole group, as well as individuals. Based on their observations, educators can adjust their instruction as needed. |
|
Provide positive and constructive feedback |
To enhance student learning, educators should provide immediate feedback while monitoring student performance. They should let students know when they have done something correctly and when they have made errors or have misperceptions. |
|
Plan for distributed practice and cumulative practice |
After students learn the content, educators can help facilitate student retention of the content and increase automaticity by implementing two types of practice:
|
Adapted from Archer & Hughes.
It is important to incorporate all components to ensure that students learn the content being taught. The thought of implementing all of these components into a lesson might seem overwhelming and time-consuming, but it is easier than it might seem, as illustrated in the videos below. The first illustrates explicit instruction being implemented during mathematics instruction at the elementary level, and the second illustrates this process at the high school level. Note: These videos are provided for illustrative purposes only to demonstrate how easy it is to implement the components of explicit instruction. They do not constitute an entire lesson and, therefore, do not necessarily reflect all 16 components. For example, these videos do not show the teacher systematically planning for instruction or implementing distributive and cumulative practice; however, we assume that she has done so by incorporating the components of those explicit instruction phases.
Elementary School Example (time: 3:34)
Transcript: Explicit Instruction: Elementary
Narrator: In this video, the teacher uses explicit instruction.
After she plans for instruction, the teacher introduces the lesson.
Teacher: Alright, boys and girls, today during math class we are going to be adding one-digit numbers by drawing pictures.
Now, in the past, we used ten frames to help us out. Show me a thumbs-up if you remember ten frames to help you out. I see lots of thumbs-up out there. Lots of you remember. We’ve also used counters before to help us out. Show me a thumbs-up if you remember using counters. I see lots more thumbs-up, too. Lots of you remember.
Well, today we’re going to be adding by drawing pictures, and we’re going to do this because you aren’t always going to have counters in your pockets or ten frames in your backpacks to help you. So today I’m going to teach you how you can draw a picture that’s going to help you add two numbers together.
Narrator: During the next phase, the teacher presents the content, leading the students through several problems while modeling the procedures and providing an example and non-example. Notice that the teacher uses clear and concise language and a brisk pace throughout the lesson.
Teacher: We’re going to start with this problem here: two plus four. To start, I’m going to draw dots to show my first number, two. One. Two. Dominique, how many dots did I draw?
Dominique: Two.
Teacher: That’s right. I drew two dots. Next, I need to draw four dots. Mateo, how many dots do I need to draw next?
Mateo: Four.
Teacher: That’s right. I need to draw four dots.
I’m going to come over here and draw four dots. Now, I want to make sure that my picture matches the problem, so I’m going to count and make sure I have one, two. And then here I have one, two…
You know, those dots are kind of messy.
If I’m going to be drawing a picture, I need my dots to be nice and neat, so I’m going to draw my dots down below… two, three, four. Now I’ve drawn four dots.
My last step is to count all the dots to see how many dots I have altogether. I have one, two, three, four, five, six dots. Carlos, how many dots do I have?
Carlos: Six.
Teacher: That’s right! I have six dots, so I know that two plus four equals six. Now, something I want you to remember: When you’re adding, sometimes you may know the answer right away, and that’s awesome. Other times, you may not know the answer right away, and that is one example of a time when you may want to draw a picture to help you add.
Narrator: After the teacher leads students through several problems, she then provides practice opportunities. During this time, the teacher offers guided practice using a variety of supports. As seen throughout the lesson, the teacher has provided frequent opportunities to respond and will continue to do so during guided practice.
Teacher: Now I’m going to have you do the next three problems with a partner. I’m going to walk around the class. I’m going to answer any questions or help you as needed.
Narrator: Throughout the lesson, the teacher has monitored the students’ performance frequently, which allows her to adjust instruction based on student understanding. She has also provided positive and constructive feedback. Once the majority of students have mastered the concept, the teacher plans for distributed and cumulative practice to facilitate maintenance. She will also identify and provide additional supports and instruction for students who have not yet mastered the concept or procedure.
High School Example (time: 5:02)
Transcript: Explicit Instruction: High School
Narrator: In this video, the teacher uses explicit instruction during a mathematics lesson. After she plans for instruction, the teacher introduces the lesson.
Teacher: Today during math class, we are going to use the tangent function to help us find the height of objects. And if you recall, this week we’ve been learning all about right triangles. Mateo, do you remember what angle makes right triangles so special?
Mateo: Ninety degrees.
Teacher: That’s right. They always contain a 90° angle. And when we have a right triangle, we know we can figure out the other angles or the lengths of the sides of the triangle using special functions. And we learned the phrase “SOH CAH TOA” to help us remember what these ratios are. Raise your hand if you remember what the S stands for. Yes, Jermaine.
Jermaine: Sine.
Teacher: That’s right. The S stands for sine. The C stands for the cosine. And, Susan, do you remember what the T stands for?
Susan: Tangent.
Teacher: That’s right. The tangent. This is what we’re going to be focused on today.
Teacher: So, using this knowledge and thinking about SOH CAH TOA to help us remember what those ratios are, we are going to solve a problem and figure out the height of a flagpole. Now, you wouldn’t normally be able to climb a flagpole or have a tape measure in your pocket at all times to help you find the height of the flagpole, so you can use one of these functions to help you figure out what the height is without having to go climb it.
Narrator: During the next phase, the teacher presents the content modeling several problems and asking questions throughout to check for understanding and to ensure student engagement. Notice that the teacher uses clear and concise language, provides an example, and maintains a brisk pace.
Teacher: So to start, I’m going to draw a picture to help me figure out what the problem’s telling me. I have a flagpole, and I know that 11 feet from the base of the flagpole is Juan.
I’m going to look back at my problem, and I notice that it says, “The angle of elevation from Juan’s feet to the top of the flagpole [so here to here] is 70°.” So I’m going to label that on my diagram. And looking back at the problem, I’ve created a diagram showing me everything the problem is telling me. But I notice something else. I notice that this flagpole and the ground make a 90° angle, which means this is a right triangle and we can use one of our ratios to help us figure out the height of the flagpole. And for this, I know I want to figure out the side opposite to the 70° angle. So looking back up there, I notice that tangent is the ratio between the side opposite and the side adjacent to my target angle, so that’s what I’m going to use. Sophie, remind me what the ratio for tangent is.
Sophie: Opposite over adjacent.
Teacher: That’s right! The tangent is the ratio of the opposite side over the adjacent side. Great thinking, Sophie. Given this equation, I’m going to then fill in all the information I have from the problem. So what is my angle in this problem? Yes.
Student: Seventy degrees.
Teacher: Great! It is 70°. So the tangent of 70° equals the opposite. I don’t know what the opposite side is, so I’m just going to leave in the word opposite over the adjacent side. I notice my side adjacent to the 70° angle is 11 feet, so I can write “11” right there. Now that my equation is written, all I have to do is solve… equals 30.25. So I know the length of the side opposite to my target angle, which is also the height of the flagpole, is 30.25 feet.
Narrator: After the teacher leads the students through several more problems, she provides practice opportunities through guided practice.
Teacher: Next, I’m going to have you work with a partner on the next two problems. Again, you’re going to be solving for the tangent function, and I’m going to be walking around, answering questions, or providing help as needed.
Narrator: During guided practice, the teacher monitors student progress and provides corrective feedback for each pair. Once a majority of students have mastered the concept, the teacher plans for distributed and cumulative practice to facilitate maintenance. She will also identify and provide additional supports and instruction for students who have not yet mastered the concept or procedure.
For Your Information
Explicit instruction is critical for teaching students effective strategies for solving mathematics problems, such as the ones presented in this module’s subsequent pages.
High-Leverage Practices
The content on this page aligns with the following HLPs.
HLP 12: Systematically design instruction toward a specific learning goal.
HLP 16: Use explicit instruction.
