Teaching Math Computation and Problem Solving Skills

Phase 1:  Pretest

During this instructional phase, a pretest is administered to each student to determine whether instruction on a particular math skill (e.g., subtraction 10 to 18) is needed.  If a student's score on the pretest is below the mastery criterion (i.e., 80%) and the student has the necessary prerequisite skills, then he or she is a good candidate for instruction on the pretested skill.  Some teachers include students who score the mastery criterion, but take more than 4 or 5 minutes to complete the 20 problems on the pretest. These students benefit from the additional practice and become more fluent.  Once the teacher has determined what instruction the students need, the teacher and students discuss the importance of learning the designated skill.  The teacher attempts to get the student's commitment to learn.  Written contracts help facilitate this process.

Phase 2: Teach Concrete Application

The concrete phase of instruction consists of Lessons 1 through 3.  For each lesson, the books provide a suggested script to guide teachers through the instructional sequence, and a "Learning Sheet" designed to facilitate student practice of the skill.  During these lessons, students learn to manipulate objects to solve the problems on their Learning Sheets.  They also begin to solve word problems in which the numbers are vertically aligned, but blank spaces are provided after the numbers for students to write the name of the manipulative object used in the lesson.  These concrete lessons act as a springboard for learning the skill at the representational and abstract levels.

Phase 3: Teach Representational Application

The representational phase of instruction consists of Lessons 4 through 6.  Again, the books provide suggested scripts to guide teachers through each lesson, and a Learning Sheet that provides students with specific practice for each lesson.  During Lessons 4 and 5, students learn to use drawings of objects to solve problems.  During Lesson 6, students learn to draw and use tallies to solve problems.  Students continue to solve word problems in which the numbers are vertically aligned, but now they fill in the blanks with the name of the representational drawing rather than the manipulative objects used in earlier lessons.  Representational lessons help students understand math skills as they progress toward abstract-level instruction.

Phase 4: Introduce the Mnemonic Strategy

The transition from representational-level to abstract-level instruction is particularly challenging for many students.  Students frequently become passive when faced with problems they perceive to be difficult (i.e., they tend to guess, depend on the teacher or peers for the answer, or quit working altogether). These same students often become active and independent learners when they master a strategy that they can use to work through problem-solving processes.  Thus, Lesson 7 (in all of the books except Place Value) introduces a mnemonic math strategy called DRAW to help students solve abstract-level problems.  Each letter of DRAW cues students to perform certain procedures.  The procedures are as follows:

1. Discover the sign.  (Students look to see what math operation to perform.)

2. Read the problem.  (Students read the problem to themselves or aloud.)

3. Answer, or draw and check.  (Students think of the answer or draw tallies to figure out the answer if they can't remember. Students check their drawing and counting to be sure their answer is correct.)

4. Write the answer.  (Students write the answer in the space provided.)

In Place Value, Lesson 7 introduces a mnemonic math strategy called FIND to help students identify the 10s and 1s in double-digit numbers. Each letter of FIND cues students to perform certain procedures. The procedures are as follows:

1. Find the columns.  (Students put their pencils between the two numbers.)

2. Insert the "T."  (Students draw a T to make a place value chart.)

3. Name the columns.  (Students write a "T" above the 10s column and an "0" above the 1s column.  They can use the word "to" to help remember what to name the columns.

4. Determine the answer.  (Now students can determine how many 10s and 1s are in the number.)

Phase 5: Teach Abstract Application

The abstract phase of instruction is presented in Lessons 8 through 10.  For each lesson, a script guides the teacher through the instructional sequence.  Again, Learning Sheets are provided to facilitate continued student practice.  During this phase, students use the DRAW or FIND strategies to solve abstract-level problems when they are unable to recall an answer. They also learn the relationships between various operations (e.g., addition and subtraction; multiplication and division) and begin to solve word problems in which the numbers still are vertically aligned but are written with the names of common objects or phrases after the numbers or in a sentence format.

Phase 6: Posttest

During this phase of instruction, a posttest is administered to students to determine whether they have acquired the basic skills and are ready to proceed to the phase of instruction designed specifically to increase fluency or speed and further develop problem-solving skills.  The mastery criterion for the post-test is 90%.  Students who score below 90% should repeat one or more of the abstract-level lessons.  Once the students have achieved 90% or higher on the posttest, the teacher informs the student that now he or she needs to increase speed and solve more challenging word problems.  The teacher discusses rationales for working on these higher level skills such as ensuring success on class and standardized tests, seat work, and homework; or making shopping easier.

Phase 7: Provide Practice to Fluency

The practice-to-fluency phase takes place in Lessons 11 through 21 or 22.  Again, each lesson features a script to guide the teacher through the instructional sequence and a Learning Sheet to facilitate student practice.  Students work on three primary skills:
(a) solving word problems that become increasingly complicated as the lessons progress
(b) increasing computation rate
(c) discriminating between problems requiring different operations.

Of the seven books in the series, four of them (Addition Facts 10 to 18, Subtraction Facts 10 to 18, Multiplication Facts 0 to 81, Division Facts 0 to 81) introduce students to the FAST DRAW strategy.  This strategy helps students set up and solve more complicated word problems. Each letter of the FAST mnemonic reminds students to perform certain procedures in order to set up the problem in a numerical format.  These procedures are as follows:

Step 1 Find what you're solving for.

Step 2 Ask yourself, "What are the parts of the problem?"

Step 3 Set up the numbers.

Step 4 Tie down the sign.

Once students have changed the word problem to a numerical format using FAST, they are able to solve the problem using DRAW.  As the lessons progress, students also learn to filter out extraneous information and to create their own word problems.

To help students increase their rate of computation, l-minute timed probes and instructional games are used.  Students also are taught to discriminate between the various operations through 1-minute probe practice.  These probes and games are discussed in greater detail in the following section.

Lesson Procedures

Give an Advance OrganizerEach lesson begins with an Advance Organizer to prepare the students for learning.  In this curriculum, the Advance Organizer serves three purposes:
(a) it connects the existing lesson to the previous lesson
(b) it identifies the target lesson skill
(c) it provides a rationale for learning the skill

Describe and Model First, the teacher demonstrates how to compute the answer for one or more problems.  During this demonstration, the teacher describes the process or "thinks aloud" as the problem is computed.  Thus, students hear what they should be thinking as they compute the problems and they see the mechanics involved in solving the problems.  The students are instructed to watch and listen.  To enhance generalization across stimulus configurations, both horizontally and vertically configured problems are used as a basis for the demonstrations.  The second procedure used during the Describe and Model section of the lessons involves demonstrating a few more problems.  This time, however, the teacher begins to involve the students by asking questions about the procedures to follow when solving the problems.  The teacher uses prompts and cues to facilitate correct responses from the students.  The teacher uses this second procedure as students begin to demonstrate understanding of the process required to answer the problems.

Conduct Guided Practice.  Guided Practice gives teachers the opportunity to instruct and support students as they move toward being able to solve problems independently.  Two procedures are used during Guided Practice.  Procedure 1 involves prompting and facilitating students' thought processes. The teacher no longer demonstrates, but instead simply asks questions that guide the students through each problem in a way that ensures success.  For example, the teacher may say, "What do we do first to solve this problem? ... Yes, we look at the first number and draw that many tallies.  What do we do next?"  In Procedure 2, the teacher instructs the students to solve the next few problems on the Learning Sheet and offers assistance to individual learners only if needed.

Conduct Independent Practice. Independent Practice is an important component of all of the lessons.  The teacher directs the students to complete six or seven problems on their own.  No assistance, prompts, or cues are provided during this part of the lessons.  Thus, teachers can tell whether students can solve the problems on their own.

Conduct Problem-Solving PracticeProblem-Solving Practice is an integral component of all lessons.  To teach students the thought processes involved in problem solving, each book incorporates a graduated sequence of word problems that become increasingly complicated as the lessons progress.

Provide FeedbackFeedback is critical to effective learning and is therefore included in all lessons.  Feedback allows the teacher to recognize and praise correct student responses, thinking patterns, and progress, thereby enhancing future responses.  Feedback also allows teachers to point out error patterns and/or incorrect math processes, and then to demonstrate how to perform the task correctly.  Students practice what has been demonstrated to ensure understanding.  The teacher encourages the students and relates positive expectations for their performance on the next lesson.  Research has shown that systematic, elaborative feedback containing these components allows students to reach skill mastery in half the instructional time otherwise required.

Miller, S. P. & Mercer, C. D.  (1997).   Teaching math computation and
problem solving:    A program that works.   Intervention in School & Clinic.
32(3), 185-191.