What evidencebased mathematics practices can teachers employ?
Page 4: Explicit, Systematic Instruction
Explicit, systematic instruction, sometimes simply referred to as explicit instruction, involves teaching a specific concept or procedure in a highly structured and carefully sequenced manner. Research has indicated that teaching mathematics in this manner is highly effective and can significantly improve a student’s ability to perform mathematical operations (e.g., adding, multiplying, finding the square root) as well as to solve word problems. This strategy has been shown to be effective across all grade levels and for diverse groups of students, including students with disabilities and ELLs. The key components of explicit, systematic instruction are highlighted in the table below.
Explicit Components  
During this highly structured instruction, the teacher:
x
maintenance In behavior assessment, term used to describe the extent to which a student’s behavior is selfsustaining over time. 

Systematic Components  
During this carefully planned and sequenced instruction, the teacher:
Sample Task Analysis Task: adding two twodigit numbers

Research Shows
 An influential metaanalysis 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 kindergarten students improved their achievement.
(Doabler, et al., 2015)
How does this practice align?
HighLeverage Practices (HLP)
 HLP12: Systematically design instruction toward a specific learning goal.
 HLP16: Use explicit instruction.
Although all students benefit from explicit, systematic instruction, students with mathematical disabilities and difficulties often require it if they are to learn foundational gradelevel skills and concepts.
Steps in an Explicit, Systematic Instruction Lesson
Orientation to the Lesson
 Teacher gains students’ attention.
 Teacher connects today’s lesson to a previously related one.
 Teacher provides students with an advance organizer, explaining why the lesson content is important as well as how it relates to real life.
 Teacher uses essential questions to assess students’ background knowledge and to activate students’ thinking.
 Teacher reviews any previously learned important vocabulary, concepts, or procedures.
Initial Instruction
 Teacher models skill or procedure, while describing the problemsolving process (i.e., uses “think alouds”).
 Teacher leads students through several problems.
 Teacher points out difficult aspects of the problems.
 Teacher continually asks students questions to check for understanding and to keep them engaged.
TeacherGuided Practice
 Students actively work to solve problems individually or in small groups while the teacher provides prompts and guidance or solves problems with the students.
 Teacher scaffolds instruction.
 Teacher monitors each student’s written work or smallgroup discussions.
 Teacher provides corrective feedback in a positive manner.
x
corrective feedback
Constructive comments provided as soon as possible following the implementation of an activity in order to help an individual improve his or her performance.
 Teacher assists students or small groups who are struggling with the skill or procedure.
 Students may discuss problems with each other.
Independent Practice
 Students complete problems independently.
 Teacher checks student performance on independent work.
 Teacher identifies students with continuing difficulty and reteaches the skills.
Maintenance
 Teacher plans for opportunities to practice the skill or concept in an ongoing manner (e.g., cumulative practice).
 Teacher identifies and provides instruction for students who need reteaching or additional practice.
Source: Bender (2009), pp. 31–32; National Center on Intensive Intervention (2016)
The videos below illustrate explicit, systematic instruction being implemented during mathematics instruction, first at the elementary level and then at the high school level.
Elementary School Example (time: 3:08)
Transcript: Explicit, Systematic Instruction: Elementary
Narrator: In this video, the teacher uses explicit, systematic instruction. During the first step of explicit, systematic instruction, the teacher readies the students for the lesson.
Teacher: All right, boys and girls, today during math class we are going to be adding onedigit 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 step, the teacher leads the students through several problems, modeling the procedures.
Teacher: We’re going to start with this problem here: 2+4. 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 all together. 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 2+4=6. 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 implements teacherguided 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: After the teacher has monitored the students during teacherguided practice and provided corrective feedback, she asks students to complete problems independently. To ensure maintenance, the teacher plans for opportunities for ongoing practice and provides instruction for students who have not mastered the concept or procedure.
High School Example (time: 4:58)
Transcript: Explicit, Systematic Instruction: High School
Narrator: In this video, the teacher uses explicit, systematic instruction during a mathematics lesson. During the first step of explicit, systematic instruction, the teacher prepares the students for 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 90degree 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 step, the teacher models several problems, asking questions throughout to check for understanding and to ensure student engagement.
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 degrees. 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 90degree 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 70degree 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 degrees. So the tangent of 70 degrees 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 70degree 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 is, which is also the height of the flagpole, is 30.25 feet.
Narrator: After the teacher leads the students through several more problems, she implements 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: After the teacher has monitored guided practice and provided corrective feedback to each pair of students, she asks the students to complete problems independently. To ensure maintenance, the teacher plans opportunities for ongoing practice and provides additional instruction for students who have not mastered the concept or procedure.
For Your Information
Explicit, systematic instruction is critical for teaching students effective strategies for solving mathematics problems, such as the ones presented in this module’s subsequent pages.