Direct Instruction: Math
What are DI Math Programs?
There are three major DI mathematics programs. These include:
- Connecting Math Concepts
- DISTAR Arithmetic
- Corrective Mathematics
Connecting Math Concepts is a comprehensive developmental mathematics program designed to teach students to compute, solve problems, and think mathematically. It includes levels A-F as well as a Bridge to F program. A pre-publication Level K program also exists for students in kindergarten. Levels A-F typically correspond to grade levels; each level of the program is meant to be taught over an entire school year.
DISTAR Arithmetic was first published in 1970 and was included in original DI research (e.g., Project Follow-Through-the largest educational experiment ever conducted). There are two levels of this program-I and II. Step-by-step procedures for solving addition, subtraction, multiplication, fractions, and problems in columns are taught in this program. Students learn to solve increasingly more complex story problems.
Corrective Mathematics includes four separate modules: Addition, Subtraction, Multiplication, and Division. In addition, there are three modules called Mathematics Modules that include Basic Fractions; Fractions, Decimals, Percents; and Ratios and Equations. Multiple modules could be completed in one school year.
For Whom are DI Math Programs Appropriate?
Connecting Math Concepts and DISTAR Arithmetic are typically seen in elementary schools. Given its developmental nature, Connecting Math Concepts is often seen in general education classrooms. DISTAR Arithmetic is most often used as a remedial program in the elementary grades. Certainly Connecting Math Concepts could be used in special education classrooms with students who need focused remediation in mathematics. One often sees DISTAR Arithmetic used with students with low incidence disabilities (e.g., moderate to severe MR) given its step-by-step procedures for even the most naïve learners and Connecting Math Concepts used with students with high incidence disabilities such as LD.
Corrective Mathematics is designed as a remedial program for students in grades 3-12. Even students beyond high school could benefit from its focused instruction in specific skill areas such as basic fractions.
What are the Key Elements of Math Instruction in DI Math Programs?
Mathematics affects our lives almost daily-from balancing our checkbooks to determining how much money we need to purchase requisite items. How best to teach mathematics skills to students is getting increased attention. The National Council of Teachers of Mathematics has articulated goals that are fairly broad in focus from teaching students to value mathematics to teaching them to reason mathematically; nonetheless, students need focused instruction to learn mathematics. DI math programs are organized around a strand-based design. They also include explicit and systematic math instruction.
Strand-based design. DI math programs are organized into strands that focus on teaching prerequisite skills to mastery before more complex skills are taught. An example scope and sequence of Connecting Math Concepts Level C is provided outlining the strands in the program; one should notice when skills are introduced, how they overlap, and how skills are reviewed over time. This type of instructional format can be contrasted with current constructivist-oriented programs that focus on spiral-based designs. In constructivist-oriented programs, prerequisites are not taught to mastery; skills are revisited over time; teachers serve as facilitators who do not typically model skills before they are practiced with students. Review is central in strand-based programs to ensure skill maintenance but only after mastery is demonstrated.
Explicit instruction. Another key feature of DI math programs is the use of explicit instruction as compared to more implicit or constructivist-oriented instruction. Explicit instruction involves teacher directed modeling, guided practice opportunities with feedback, and independent practice. This independent practice is distributed over time. Again, this can be contrasted with constructivist approaches where teachers are facilitators, guiding students through authentic math activities. Students explore and discover unique ways to solve problems on their own. An exercise from Connecting Math Concepts Level C is provided to show the teacher-directed nature of the program. Exploration and discovery are down-played; it is much more efficient to show students how to perform a skill and then to give them practice opportunities to ensure success.
What do Sample Lessons from DI Math Programs Look Like?
Lesson 108 from Connecting Math Concepts Level B is provided. This lesson illustrates how strands are incorporated into the lesson and how explicit instruction is utilized by teachers.
Lesson 30 from Corrective Mathematics: Subtraction is illustrated. This lesson provides focused and explicit instruction in subtraction and includes multiple opportunities to practice acquired skills.
What Format Features Make DI Math Programs Unique and How Can We Use these Features without having These Programs Available?
There are several format features that make DI math programs unique. These include: (a) clear teacher scripts, (b) placement tests and within program assessments, (c) choral (unison) responding and signals, (d) individual turns, and (e) error correction and verification techniques. If teachers do not have access to DI math programs, they can still use these format features to enhance other published programs or teacher-developed lessons in the classroom.
Clear teacher scripts. Clear teacher scripts specify what teachers say (typically noted in color) and do (noted in regular print) and what students say or do (noted in italics).Corrective Mathematics is different in this regard-regular type illustrates what teachers say; italics illustrates what teachers do; and bold print illustrates what students say.
Words (or numbers) noted in bold in the teacher script are referred to as "pause and punch" words. These words should receive increased emphasis by the teacher (e.g., are said louder by the teacher or are said after a pause to stress the word). An example script is shown from Lesson 63 of Connecting Math Concepts Level A.
If teachers do not have access to DI math programs, they may use scripts such as these to ensure consistency in the delivery of instruction in the classroom. Instructional assistants, parent volunteers, and tutors would benefit from having clearly defined instructions to provide to students. Substitute teachers would also benefit from clear scripts.
Placement tests and within program assessments. Placement tests and within program assessments allow teachers to place students in the proper level/lesson of DI language programs and to track their performance over time. Students can be skilled grouped based on their performance on the placement test. The placement test from DISTAR Arithmetic I is illustrated.
Within program assessments help determine the efficacy of instruction. Teachers may decide to repeat lessons to ensure firm responding (mastery) before moving on or accelerate students to higher lessons/levels based on their performance.
Within program assessments occur every day or at regimented points in the program. They include workbook activities and mastery tests. The sample lessons from Connecting Math Concepts and Corrective Mathematics: Subtraction provided above illustrate workbook activities conducted in the programs.
If teachers do not have access to DI math programs, they can still incorporate aspects of assessment into their daily teaching. For example, they can survey what skills will be taught to students during the upcoming year and assess whether students have these skills or not on a teacher-developed pretest. Further, as teachers provide instruction in the classroom, they can assess key aspects of the lesson to determine if further instruction is needed or if they can proceed to the next lesson. Assessment informs instructional practice and should be used in some capacity. DI math programs make it easy for teachers to assess students given that everything is in place-teachers do not have to develop anything. They just need to analyze performance and make data-based decisions.
Choral (unison) responding and signals. Choral or unison responding increases students' opportunities to respond and to receive teacher feedback. It is far better to have students respond in unison when they are first learning a skill than to call on students one at a time to respond. When choral responding is utilized, students should respond together (like one person said it), respond correctly, and "say it like they know it," responding on signal (described next).
Signals are used to prompt students to respond together. If we have students echo one another, they may or may not have acquired the desired skill (they may be simply listening to others). The key to signals is to remember the following:
- If students' eyes are on the teacher, use a hand drop signal. For example, "Say 15 and count backward to 5. Get ready." The teacher should have her hand up in the stop position when she is talking and drop her hand when she wants students to respond.
- If students' eyes are on not on the teacher (they are looking at their textbooks, for example), the teacher should use an audible signal. Audible signals include finger snaps, taps with the pencil, or claps that are stated after the teacher provides the directive
- If students are looking at the board, the teacher should use a point-touch signal. For example, if the teacher points to a number line and says, "you're going to tell me the number that is 2 less than the number next to the arrow card," she would tap the board under the arrow card (it covers the answer), evoking a unison oral response from the students.
If DI math programs are not available, teachers can still use choral responding in the classroom. Saying, "Everybody, what is 2 + 2?" can prompt all students to come in together. The teacher may also use various signals in the classroom to evoke student response such as the hand drop, point touch, or audible signals.
Individual turns. Following choral (unison) responses, teachers should ask for individual turns. The rule of thumb in DI language programs is 85% group responses followed by 15% individual responses. Before individual turns are provided, teachers should ensure that group responding is firm. That is, students should "say it like they know it." Once this is evident and group responses have been performed on a task, the teacher can announce, "time for turns." When time for turns is used, teachers should always use a student's name at the end of the directive. For example, "What is 4 x 4, Bill?" would be used as compared to "Bill, what is 4 x 4?" In this way, all students are attentive and ready to work! The following example from DISTAR Arithmetic illustrates the use of individual turns (noted at the bottom of the format).
If DI math programs are not available, individual turns can still be utilized in the classroom. Again, they should occur after choral responses are provided and when the group is firm (shows mastery).
Error correction and verification techniques. DI math programs have specified error correction techniques. An example from Corrective Mathematics: Addition is provided.
As can be seen, the error correction includes a teacher model, an opportunity for students to perform the task, and a review of the item (called a delayed test or starting over).
If teachers do not have access to DI math programs, the following error correction procedure can be used for most mistakes.
(Show students how to do it).
Do it with me.
(Show students how to do it by doing it with students).
(Have students do it on their own).
(Do a starting over; start over at the beginning of the activity to ensure that students can demonstrate the correct response).
For example, the teacher says, "How many tens in 30? Students respond, "2." The error correction would look like the following:
- My turn. How many tens in 30? Three.
Your turn. How many tens in 30?
Repeat task after doing several other problems. "How many tens in 30?"
The rule of thumb is to provide a model followed by student practice. A "starting over" should be incorporated as well to ensure that students "have it."
In addition to error corrections, teachers should also be liberal in their amount of praise or verification statements provided. A rule of thumb is to say, "yes" plus whatever the students said. Teachers can use this strategy with or without the use of DI math programs. For example, when students respond, "16" to the instruction, "What is 8 x 2?" the teacher could say, "Yes. 8 x 2 = 16."