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Seni Belajar dan Mengajar Matematika untuk Tunarungu

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Use Technology
Computers:   Math software games are a good way to incorporate learning and students will enjoy these games more than drills or paper and pencil worksheets.  Most software relies heavily on audition and/or reading.  Teachers can preview games with D/HH learners to help them understand the "game" and to explain the logistics of the software.  After a few teacher-mediated sessions with a piece of software, the game (and more importantly the math practice) is often accessible to D/HH students.  If needed, a script can be kept near the computer to help the student with directions to particular sections of the game.
Calculators:  Calculators are an integral part of problem solving and written communication.  They should be used for problems that are complex or require multiple steps.   Students should be able to utilize calculators when they cannot work out problems themselves.
Approach and Attack
As students approach a math problem, they should be encouraged to:
·         Restate the problem in their own words.
·         Make a personal plan for solving the problem
·         Try out their plan.
·         Check to see if their answer makes sense and if it solves the problem.
·         Describe (orally or in writing) how they reached this solution.
Making Math Visual     
1.       Draw a picture of the problem.
2.       Make a chart or diagram.
3.       Act it out or use manipulatives.
4.       Create a mental picture of the  problem.
Writing and Problem Solving
Lexical items often have a unique meaning in mathematics.  To help students with words like this (e.g., ray) students should keep a personal glossary.   Have students keep a small notebook with new  vocabulary, student-drawn diagrams, and short descriptions if necessary.  
 To help both the teacher and students "see" how a problem has been solved, have students fold a paper vertically down the middle.  On the left, the students should record their problem solving.  On the right, they should write the explanations of how they solved the problem.

Enhancing the Teaching of Mathematics
Start Early and Maintain Standards- Strong mathematics education must begin in the early grades, be provided daily, and be offered throughout a student's education.
More Mathematics in High School- D/HH students, and especially those expecting to attend college, should be given the benefit of four full years of mathematics at the high school level.
Attitudes Matter- Teacher attitudes can and will have an impact on students.  Students must be shown the value of mathematics in their daily lives.
Technology Helps- Scientific and graphing calculators are widely available and teachers must make use of these in their classroom.
Language is Important- Language that supports mathematical thought needs to be developed, expanded, and practiced throughout the school day.
Get Support- Teachers should be aware of the different resources available that provide strategies and classroom activities.  
"Space it Out"
1.       Distributed practice:  Practice often for short periods of time, rather than devote an hour every other day.
2.       Emphasize "reverses" or "turnarounds":  When teaching 4+5, teach 5+4 also.
3.       Group facts in small chunks and teach to mastery.   After students master several tasks, mix them up and test for comprehension.
Make Math Meaningful
Instructional activities should be designed to relate math to the student's own experiences.  Below are strategies with an example for each.
1.       Select tasks to keep students interested and intellectually challenged.
-          Teach percentages in relation to tax or a popular video game.
-          Ask students to calculate the percentage of their paycheck that is withheld for taxes.
2.       Help students use math tools to investigate problems.
-          Use protractors, rulers, or yard sticks when approaching problems of length in geometry. 
3.       Make connections to prior knowledge and developing skills.
-          Collaborate with students' content area teachers (or try thematic units in your own curriculum) so that mathematical problems are part of other subject areas.  
Vocabulary of Mathematics
To help deaf children learn the vocabulary of mathematics, the teacher has to search for the point of contact that will permit the child to add new information to his or her knowledge base.  One sequence of instruction is as follows:
1.       Identify the strengths and interests of the child and use this information to motivate the students.
2.       Expand the current knowledge base through broadening experiences.
3.       Teach the concept in relation to known information.
4.       Teach the vocabulary, which includes some or all of the following: spelling, fingerspelling, sign, and pronunciation.
5.       Reinforce the meaning (and multiple meanings) of the vocabulary in other nonmathematical activities.
A number of techniques at hand to help teach the meaning of both common and specialized mathematics vocabulary:
1.       Incorporate the teaching of the word in a language arts activity such as spelling or writing a story.
2.       Use mathematical terms in notes that you write to a student.
3.       Write the words on cards and place them on the walls with examples of their meaning.
4.       Have the students compile a list of key mathematical words by entering them into a vocabulary book along with a description and an example of their meaning.
5.       Make a point of using mathematical terms in your own dialogue.
6.       If your students sign, then make ample use of fingerspelling to clearly indicate which mathematical term you are using.
7.       Have the students explain the meaning of mathematical terms to the class.
8.       Teach deaf students how to talk about learning mathematics.
Challenge Students to Describe and Analyze their Solution Methods
An important step in advancing children's thinking is to challenge them to describe and analyze their solution methods.  The following paragraphs illustrate instructional strategies that Ms. Smith used to elicit children's solution methods
Elicit many solution methods for one problem.  Rather than focus on a single answer to a mathematics problem, Ms. Smith attempted to foster discussion about how students solved a problem.  She asked such questions as "Who did this problem another way?", "Did anyone do something else, that Allan did not do?" and "Can you use anything else besides your fingers and a number line to solve this problem?"  By asking these types of questions, Ms. Smith encouraged children to share their ideas.  Moreover, children in this classroom readily discovered that many approaches are available to solve problems.
Wait for, and listen to, students' descriptions of solution methods.   A sense of calm and patience is needed to encourage children to express their ideas.  Providing sufficient wait-time and listening to ideas lets children know that thoughtful explanations are more valuable than quick answers.
Encourage elaboration.  Often, children need prompting to explain their thinking more completely.  Even though Ms. Smith may have understood a child's response, she encouraged clarifications for the benefit of the entire class. On occasion, she assisted students in articulating their methods.
Use students' explanations for the lesson's content.  Students' articulated ideas can furnish the content of class discussions.
Remind students of conceptually similar problem situations.  To "jump-start" their thinking, children may need to be reminded that one problem is like another one that they have solved previously.
Review background knowledge.  Reviewing necessary background knowledge with students is another effective support strategy.  For example, Ms. Smith reviewed coin values for a student who was having trouble counting money.
Lead students through instant replays.  Teachers can support the understanding of all children in the class by revisiting one child's elicited solution method in a slow and step-by-step fashion.  This strategy is very different from that of a teacher who offers his or her own solution method as the only sanctioned method.
Write symbolic representations of each solution method on the board. Writing the symbols for the children's solution methods on the board helps children link the verbal descriptions of their thinking with the written mathematical marks.  Ms. Smith noted another benefit of this strategy: "Recording on the board assists students in following the procedure.  Some students must see the numbers.  Constant review of this helps them write the digits."
Encourage students to request assistance.  Ms. Smith expected children to request additional help when necessary.  She attached no stigma to the requirement for extra help; on the contrary, students who requested assistance received extra time and attention from the teacher.  This acceptance and expectation is an important aspect of a teacher's support of students' learning
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 Organizer.  Each 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 Practice.  Problem-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 Feedback.  Feedback 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.
Involve Students in the Assessment Process
    The Curriculum and Evaluation Standards established by the NCTM calls for teachers to link assessment to instruction and involve students in the assessment process to gain insights into students' knowledge of math and the ways in which students think about mathematics (NCTM, 1989).  Therefore, assessment should include student-centered strategies that involve students in setting goals, choosing appropriate assessment techniques, and identifying appropriate instructional strategies and materials.
Use Peer-Mediated Instruction
Peer-mediated instruction can encourage the development of classrooms as mathematical communities by allowing students to work together to communicate about and experiment with their own solutions to mathematical situations (Buchanan & Helman, 1993; NCTM, 1989, 1991; Scheid, 1994).  Rather than being teacher-directed, peer-mediated learning arrangements provide students with the opportunities to work in groups to formulate and pose questions; share ideas; clarify thoughts; and experiment, brainstorm, and present solutions with peers (Lo, Wheatley, & Smith, 1994).
Successful peer-mediated instructional strategies that help make math an interactive problem-solving experience include peer tutoring (Scruggs & Richter, 1985), class-wide peer tutoring (Maheady, Sacca, & Harper, 1987), reciprocal peer tutoring (Fantuzzo, King, & Heller, 1992), team-assisted individualization (Slavin, Madden, & Leavey, 1984), and Learning Together (Johnson & Johnson, 1986).
Suggestions for Implementing Successful Peer-Mediated Instructional Arrangements:
1.       Establish rules and guidelines to help students function as a group and work cooperatively (Johnson & Johnson, 1990)
2.       Form heterogeneous groups based on such factors as gender, race, ethnicity, linguistic ability, academic and social level, and ability to work together (Dishon & O'Leary, 1991; Edwards & Stout, 1990)
3.       Arrang the physical design of the classroom to facilitate peer-mediated instruction (Johnson & Johnson, 1986)
4.       Help students learn to work cooperatively (Johnson & Johnson, 1990; Whittaker, 1991)
5.       Deal with problems that typically occur with peer-mediated instructional arrangements, such as increased noise levels, complaints about partners, and cheating (Maheady, Harper, & Mallette, 1991)

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