Masalah Difabel
Seni Belajar dan Mengajar Matematika untuk Tunarungu
http://deafed.department.tcnj.edu/math/
TIPS UNTUK SISWA
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.
(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
(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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