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1. Using calculator in Mathematics in Elementary level:
1 .The 'battle' over calculator use
Some people say calculator enables children to concentrate on understanding and studying
mathematical concepts instead of spending time on tedious calculations. They say calculator helps
develop number sense, and makes students more confident about their math abilities
National Council of Teachers of Mathematics (1989) has recommended that long division and
"practicing tedious pencil-and-paper computations" receive decreased attention in schools, and that
calculators be available to all students at all times.
Others are against using calculator in lower level math teaching, saying that it makes children not to
learn their basic facts, prevents students from discovering and understanding underlying mathematical
concepts and instead encourages them to randomly try different operations without understanding
what they're doing.
They say calculators keep students from benefiting from one of the most important reasons for
learning math -- to train and discipline the mind and to promote logical reasoning.
2. Advantages using Calculator
In my opinion, calculator can be used in the teaching in a good way or a bad way - it all depends on the
teacher's approach. Calculator in itself is not bad or good -- it is just a tool. It is used a lot in today's
society, so students should learn to use it by the time they finish school.
At the same time, children SHOULD learn their basic facts, be able to do mental calculations, and
master long division and other basic paper-pencil algorithms. Mathematics is a field of study that builds
on previously established facts. A child that does not know basic multiplication (and division) facts will
have hard time learning factoring, primes, fraction simplification and other fraction operations,
distributive property, etc etc. Basic algorithms of arithmetic are a needful basis for understanding the
corresponding operations with polynomials in algebra. Mastering long division precedes understanding
how fractions correspond to the repeating infinite (non-terminating) decimals, which then paves way to
understanding irrational numbers and real numbers. It all connects together!
For this reason, it is probably very wise to restrict the calculator use in the lower grades, until a child
knows her basic facts and can add, subtract, multiply, and divide even large numbers with pencil &
paper. THIS, in my opinion, can build number sense — as do mental calculations.
This does not mean that you couldn't use calculator occasionally in the elementary grades for special
projects or when teaching specific concepts, or for some fun. It could be used for example in science or
geography projects, or for exploring certain new concepts, or for some number games or checking
homework. See below for some ideas.

2. The discussion here does not apply to graphical calculators in high school. I am strongly in favor of
using graphical calculators or a graphing software when studying graphing of functions and calculus.
Even there though, one certainly needs to learn the basis of how the graphing is done on paper.
3. Things to keep in mind when using calculator
When calculator is used more freely, one should pay attention to following points:
Calculator is a tool to do calculations. So is the human mind, and paper & pencil. Children should be
taught when to use calculator, and when mental computing (or even paper & pencil) are more effective
or appropriate. Choosing the right 'tool' is part of effective problem-solving process.
It is very important that students learn how to estimate the result before doing the calculation. It is so
very easy to make mistakes when punching in the numbers and a student must not learn to 'rely' on the
calculator without checking the reasonableness of the answer.
Calculator should not be used for a random trying out of all possible operations and seeing which one
produces the right answer. It is crucial that the child understands the different mathematical operations
so she knows WHEN to use which one - whether the actual calculation is done mentally, on paper, or
with a calculator.
4. Ideas for calculator use in elementary grade math
If you use these ideas, make sure the kids don't get the idea that calculator takes away the need to
learn mental math. It can serve as a tool to let children explore and observe, but afterwards the teacher
should explain things, justify the math rules, and put it all together.
Let preschoolers or first graders explore numbers by adding 1 repeatedly (which can be done with first
punching in 1 + 1 =, and then pressing the = button repeatedly) or subtracting 1 repeatedly. Observe
their faces when they hit negative numbers! Or, let them investigate what happens to a number when
you add zero to it.
Calculator pattern puzzles: An extension of the idea above, where first-third grade children add or
subtract the same number repeatedly using a calculator. Children will observe patterns that emerge
when you add 2 or 5 or 10 or 100 repeatedly, or will make their own "pattern puzzles" which are simply
number sequences with a pattern where you omit some numbers, for example 7, 14, __, __, 35, __, 49.
The activity can connect with the idea of multiplication very easily.
Place value activity with calculator: Students build numbers with the calculator, for example:
Make a three-digit number with a 6 in the tens place; OR Make a four-digit number larger than 3,500
with a four in the ones place; OR Make a four-digit number with a 3 in the tens and a 9 in the hundreds
place; etc.

3. Write number one million on the board. Ask students to pick a number that they will add repeatedly
with the calculator to reach one million within reasonable class time. If they pick small numbers like 68
or 125 they won't reach it! This can teach children about how vast and big the number one million is.
When introducing pi, have students measure the circumference and the diameter of several circular
objects, and calculate their ratio with a calculator (which saves time and can help keep the focus on the
concept).
Cheating in Mathematics subject:
 Academic cheating is defined as representing someone else's work as your own. It can take
many forms, including sharing another's work, purchasing a term paper or test questions in
advance, paying another to do the work for you.
 In the past it was the struggling student who was more likely to cheat just to get by. Today it is
also the above-average elementary bound students who are cheating.
 Cheating no longer carries the stigma that it used to. Less social disapproval coupled with
increased competition for admission into universities and graduate schools has made students
more willing to do whatever it takes to get the A.
 Grades, rather than education, have become the major focus of many students.
 Many students feel that their individual honesty in academic endeavors will not affect anyone
else.
 Students who cheat often feel justified in what they are doing. They cheat because they see
others cheat and they think they will be unfairly disadvantaged. The cheaters are getting 100
on the exam, while the non-cheaters may only get 90's.
 In most cases cheaters don't get caught. If caught, they seldom are punished severely, if at all.
 Cheating increases due to pressure for high grades
 Math and Science are the courses in which cheating most often occurs.
 Computers can make cheating easier than ever before. For example, students can download
term papers from the World Wide Web.
 Cheating may begin in elementary school when children break or bend the rules to win
competitive games against classmates. It peaks during high school when about 75% of students
admit to some sort of academic misgivings.
 Research about cheating among elementary age children has shown that: There are more
opportunities and motivations to cheat than in preschool; Young children believe that it is
wrong, but could be acceptable depending on the task; Do not believe that it is common; Hard
to resist when others suggest breaking rules; Need for approval is related to cheating; Boys
cheat more.
 Research about cheating among middle school children (Ages 12-14) has shown that: There is
increased motivation to cheat because there is more emphasis on grades; Even those students
who say it is wrong, cheat; If the goal is to get a good grade, they will cheat.
 Academic cheating begins to set in at the junior high level.

4.  Cheating is seen by many students as a means to a profitable end.
 Cheating does not end at graduation. For example, resume fraud is a serious issue for
employers concerned about the level of integrity of new employees.
Best strategies for teaching elementary math:
Elementary math is a subject many students struggle to grasp. The material often requires
extra attention and differs with each student. According to Best Evidence Encyclopedia, there is
a large achievement gap between black, Hispanic and white elementary students. Here are
some of the best and most effective methods of teaching math which may work to close the
learning gap and help those who often experience learning hurdles in the classroom.
Visuals and graphics
Textbooks often include various visuals and graphics for students to learn from. They are
crucial elements to accompany text and help get the concepts across to students. However,
according to the National Council of Teachers of Mathematics, these graphics appeared to be
much more effective when paired with specific practice or guidance. This guidance could be
coming from the teacher or from another classroom tool.
Teachers are starting to implement computerized learning into the classroom for a more
personalized learning approach. In fact, more than four in 10 teachers report the use of e-readers
and tablet computers in their classrooms to complete assignments and assist in
learning. With virtual math programs, children can not only see these graphics, but they can
revisit concepts that were especially difficult. They are able to learn at their own pace and
won't feel rushed to move onto a concept they are not yet ready to tackle.
Verbalized thinking
The process of having students verbalize step-by-step how they got to the answer they did
may help other students to learn basic procedures. As you may know, many students are
hesitant to raise their hands in class to ask questions for fear of sounding unintelligent. If those
in the class are required to explain how they got to that answer, they may be helping their
classmates in the process. The act of students explaining their process may also help them to
learn how to do a certain problem. This helps them to recognize the strategies they're using
and potentially apply them to other areas of learning.

5. Specific feedback
According to the Institute of Education Sciences, many students benefit from specific teacher
feedback about what they did correctly and where and how they can improve next time.
Teachers should also present their students with opportunities to correct their answers and
see what errors they made. Instead of simply giving the correct answer and telling them where
they went wrong, it helps for students to be guided in that direction so they can figure it out
for themselves.
New Teacher?
Beginning your journey as a mathematics teacher? Empowering the Beginning Teacher of
Mathematics in Elementary School has been created to help you reach your full potential as a
mathematics educator. Resources cover professional growth, curriculum and instruction,
classroom-level assessment, classroom management and organization, equity, and school and
community.
Representation a Model for Understanding, Using, and Connecting
Representations:
Teachers can reflect on their practice by examining a model for representation. Students’
thinking about problem solving as reflected in these representations may differ from their
teachers’ thinking. The Representations Model provides a lens for making sense of students’
responses to tasks.