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Perhaps the earliest device for working out sums was the abacus. This began as a clay tablet into
which grooves were cut. Pebbles were then placed or taken away from grooves to perform addition
and subtraction. (Our word calculation comes from the Latin word calculus meaning 'pebble').
Because the pebbles were likely to become mislaid, they were later replaced by beads threaded on to
wires and mounted in a frame. By moving the beads backwards and forwards, addition, subtraction,
division and multiplication could be done. In 1614 John Napier, an astronomer, invented a ready–
reckoned, known as Napier's bones, to help him make complex calculations accurately. From this
was developed in 1621 the earliest form of the slide rule.
The first ... Show more content on Helpwriting.net ...
These circuits are mainly used for solving a wide variety of complex engineering problems, like
investigation of stresses in aircraft, ships and large engineering structures. They can also be used to
simulate and set up models of complex installations and study the effect of various operational
factors on the complex installations.
They can for example, be used to simulate the behavior of an aircraft in response to the actions of
crew members. Using analog computers, apparent equipment failures or other emergencies can be
introduced for proper training of the crew. The first automatic analog computer designed to solve
complex differential equations, was described in detail in 1876 by the English Scientist William
Thomson.
Digital Computers: These are used in commerce and industry for extensive arithmetical calculations
which would otherwise require enormous clerical effort. Such computers carry out mathematical
operations with the variables expressed in the computer as numbers, usually in the binary system
(given below).
These numbers are recorded in the computer electronically, as a series of temporary magnets, each
magnetized in one of the two possible directions. The two magnetization direction corresponds two
numbers of the binary system.
The first electronic digital computer, known as the Electronic Numerical Integrator and Computer or
NIIAC was developed at the University of Pennsylvania in 1947 by
... Get more on HelpWriting.net ...

Children 's Knowledge And Understanding Of Place Value
EDP243 Children as Mathematical Learners Assessment One Child Study – Diagnosing Children's
Knowledge and Understanding of Place Value Tayhla Wood 18866371 Tutor: Kerrie Maxwell The
following child study is divided into two components discussing aspects of place value as a concept
taught in mathematics education. Component A introduces the topic by discussing the significance
of interview style assessment, followed by discussion on the importance and teaching of place
value. Misconceptions and difficulties surrounding place value learning are also described.
Component B focuses on an individuals understanding of place value through the use of an
interview and tutoring sessions. 1. COMPONENT A 1.1 Rationale Essential aspects that underpin
the professional and dedicated educator include the revising of knowledge and experience,
reflection, and an effort in understanding their students. Within mathematics, these skills are
informed by the curriculum chosen, the students involved, and the pedagogy that is selected, that
create the professional judgement cycle (as seen in Appendix One) (Department of Education and
Training Western Australia [DETWA], 2013a). The more teachers understand about their students,
the more they can adapt the learning environment to cater for these different learning approaches
(Burns, 2010). Assessing is a crucial component of education that informs teachers on individual
development and understanding (Booker, Bond, Sparrow

History Of Roman Number Symbols
The history of Roman Number Symbol are represented by letters. The Roman numerbs are
represented by seven different letters are I, V, X, L, and D. Therefore, these roman letters represent
1, 5, 10, 100, and 500. Ancient Roman use these seven letters to make a lot of different numbers and
to be written of the Roman alphabet. In the Etruscans was an ancient civilization of Italy developed
their own numeral system with different symbols. A common theory of the origin Roman numeral
system was represented by hand signal. For example, the Roman numeral system by hand signal
was used like one, two, three and four signaled by the equivalent amount of fingers that were used.
Then, "The number five is represented by the thumb and fingers separated, making a 'V' shape and
The number ten is represented by either crossing the thumbs or hands, signaling an 'X' shape"
(Pollard). Therefore, the numbers; six, seven, eight and nine are represented by one hand signaling a
five and the other representing the number 1 through to 4. The hand signal was used for counting by
either crossing the thumbs, fingers separated, and signaled, which helped to hand ... Show more
content on Helpwriting.net ...
The tally sticks had been used for thousands of years and continued to be used until the 19th
century. For instance, the tally sticks was used to either additive nor subtractive, whereas the
numbers one, two, three and four were represented by the equivalent amount of vertical lines. If
these numbers described in the article would be written in tally sticks a Roman numerals. For
instance, "Four could be written as either IIII or IV" (Reddy). Another example, seven on a tally
stick would look like, IIIIVII, when shortened it would look like VII. These Roman numbers are the
same like the Roman number symbol. Another reason, larger numbers in tally sticks like 500 and
1000 would be a 'D' and 'M' in a circle

The History of Zero: Indian and Mayan Cultures
Zero is usually recognized today as being originated in two geographically separated cultures: the
Maya and Indian. If zero was a place–holder symbol, then such a zero was present in the
Babylonian positional number system before the first recorded occurrence of the Indian zero. If zero
was represented by an empty space within a well–defined positional number system, such a zero
was present in Chinese mathematics a few centuries before the beginning of the Common Era. The
absence of a symbol for zero in China did not prevent it from being an efficient computational tool
that could handle solution of higher degree order equations involving fractions.
However, the Indian zero was a symbol, a number, a magnitude, a direction separator and a place–
holder, all in one operating within a fully established positional numeration system. Such a zero
occurred only twice in history – the Indian zero which is now the universal zero and the Mayan zero
which occurred in solitary isolation in Central America at the beginning of the Common Era. To
understand the first appearances of the Indian and Mayan zeroes, it is necessary to examine them
both within the social contexts in which both of these inventions occurred. Because of the popular
difficulties with the zero, there has occurred over time a series of avoidance mechanisms to cope
with the presence of zero. The word zero comes from the Arabic meaning void or empty which
became later the term for zero. The ancient Egyptians never

Essay On The Babylonians
The Babylonian civilization existed from around 3000 BC until 539 BC. The civilization has its
roots to Mesopotamia, a plain between the Tigris River and the Euphrates River, which is now in
modern day Iraq. The Babylonians are notoriously famous in their discoveries and inventions, most
of which we still use today. The civilization–developed ideas such as astronomy, in which they
created the Saros Cycle, a cycle used to predict solar and lunar eclipses. The Babylonians were also
masters in architecture, constructing buildings like the Hanging Gardens using innovative
techniques including glass–covered bricks and adornments of gold and bronze. However, despite
many of these discoveries being forgotten or lost, their advancements in mathematics have been
kept for centuries, and have brought modern ideas of angles, quadratic equations and even
knowledge about the Pythagoras theorem, before Pythagoras himself existed.
One modern–day use of mathematics and quadratic equations, created by Babylonians, was the
collection of taxes. Mesopotamia was a very fertile land; therefore there were a high number of
farmers in the Babylonian Empire. Because of the high number of farmers, tax collectors and
mathematicians had to develop a new and more efficient way of calculating the ... Show more
Despite the fact that many of their discoveries have been lost through time, it can be concluded that
the Babylonian Empire was very advanced for its time and despite ending with a societal collapse, it
is still possible to find everyday things without knowing they possess century–old roots connecting
to the ancient Babylonian civilization. This truly shows, the large influence the Babylonian
civilization has on our modern day society and our understanding of the

HOW CHILDREN UTILIZE THEIR MATHEMATICAL MIND AS
PART OF...
"Dr Maria Montessori took this idea that the human has a mathematical mind from a French
philosopher Pascal and developed a revolutionary math learning material for children as young as 3
years old. Her mathematical materials allow the children to begin their mathematical journey from a
concrete concept to abstract idea". With reference to the above statement please discuss how these
children utilize their mathematical mind as part of their natural progression, to reason, to calculate
and estimate with these Montessori mathematical materials in conjunction with their aims and
presentations? What is a mathematical mind? The Mathematical Mind' refers to the unique
tendencies of the human mind. The French philosopher Blaise Pascal said ... Show more content on
Helpwriting.net ...
A significant discovery that Dr. Montessori made was the importance of offering indirect
preparation for the math materials while children were in the sensitive periods for movement and
the refinement of the senses. It is through children's work with the Exercises of Practical Life and
Sensorial materials that they first encounter and experience the concepts of measurement, sequence,
exactness, and calculation Sensorial education is the basis of mathematics. Dr. Montessori said that
children are sensorial learners. They learn and experience the world through their five senses. So
sensorial education helps the child to create a mental order of the concepts he grasps using his five
senses. "The skill of man's hand is bound up with the development of his mind, and in the light of
history we see it connected with the development of civilization." – Maria Montessori, THE
ABSORBENT MIND, Chap 14. pg. 138 Montessori firmly believed that the 'hands' are the mother
of skills. By providing Montessori sensorial materials to the child she was convinced that correct
manipulation with quality and quantity would certainly create a lasting impression in the child's
mind with the understanding of mathematics. We place materials quite intentionally on trays, we
color code activities, materials are displayed

The Importance Of Student Achievement
Student achievement is a multi–faceted concept and does not look the same for every student. At the
core of student achievement are comprehension and connection. Students must be able to
understand and contextualize instructional material to foster comprehension. Equal to
comprehension is the ability of students to connect with the material and make relevant associations
between concepts and skills for application. The intention of instruction based on prioritizing and
unwrapping standards helps educators to define the instructional pathway that best supports the
standard and facilitates student learning. Learning pathways align essential objectives, activities and
resources to outline progress within a standard. Learning pathways detail the skills students need to
acquire for demonstrating comprehension and mastery, which constitute achievement. Additionally,
learning pathways present an interpretive value for gauging student understanding and performance.
Gutierrez explains that learning pathways "must have strategic opportunities to build in prior
knowledge, reflection, and application" (Gutierrez, 2017, para 8). Gutierrez further positions that
"learning pathways are intended to be flexible, multidisciplinary, and increasingly personalized"
(Gutierrez, 2017, para 8). Without properly considering and using learning pathways, students are
not provided opportunities to connect content with comprehension and skill. Effective learning
pathways scaffold on what students

Common Sense Abbreviations
The common sense approach to abbreviations should apply to PSI reports as well. Of course one can
abbreviate suffixes (Dr. Mr., etc.), words like rd. (road) st. (street) and ave. (avenue). Agency names
could be spelled out completely and followed in parenthesis, i.e. State Bureau of Investigation
(SBI). Beyond these, I see no reason to include abbreviations to the point that they are even a topic
of discussion. Probation officers are not really relaying a personal opinion in the PSI. They present
documented and measurable progress of the individual under supervision. The officer might have
assigned a rating for the offender based on their behavior during an interview, such as level of
remorse on a Likert scale. The recommendations

Questions On Learning And Teaching
Task #4: Mathematics Learning & Teaching
Competency 662.1.4: Aligning Learning Activities to National Standards
Competency 662.1.5: Standards and Best Practices in Teaching and Learning
Competency 662.1.7: Differentiated Instruction
Jennifer Moore
Western Governor's University
Part A: The "Equivalent Fractions and Decimals Lesson Plan" is aligned to NCTM's content and
process standards. The content standard that this lesson is addressing is numbers and operations.
This entire lesson is about students using fractions and decimals to solve problems. This lesson also
has several process standards addressed in the lesson plan. One of the process standards used in this
lesson is Connections. Throughout this ... Show more content on Helpwriting.net ...
"The Show or the Ad" activity has students using word problems that talk about television programs
and commercials. The teacher has made the students connect that math is even in the TV shows that
they watch. The teacher then makes it even more the students own connections by allowing them to
be creative and writing their own word problem about TV and commercials. The teacher also uses
these types of problems on the summative assessment. The teacher uses connections even in the
homework that is assigned to the students. The teacher assigns students to watch their favorite
television show, and as they watch the show to record how many minutes the commercials are. All
of this shows the students understanding of the concept while allowing the students to relate this to
something they enjoy. The one NCTM process standard that I would add to this lesson plan would
be the reasoning and proof standard. This is the only standard that is rarely used in this lesson plan.
"The Reasoning and Proof standard emphasizes the logical thinking that helps us decide if and why
our answers make sense" (Walle, p. 5). This lesson needs to allow students to justify the reason for
their answers. This lesson only has them justifying their answers on one question, throughout the
entire lesson. I would add more to the questions than just the answers. One part of this lesson I
would change to incorporate the reasoning and proof standard is on the "The Show or the

Examples Of Josephus Problem
The next base the essay will explore is base 3 also known as ternary, this is the base that was used in
the original Josephus problem. By exploring this base and a few more bases, I will find a pattern and
derive an equation to solve Josephus problem with any number of people in any number base.
Josephus problem in base 3 is when each person kills the person who sits two seats away from him
and the following diagram will demonstrate how it happens, the numbers on the inside show the
order of the people that are gone which will make it easier to follow.
Figure 7 An example of how Josephus problem works in base 3 with 12 people
I will now create the same table as I did for the problem in a binary form.
Table 3 Analyzing the Josephus problem in base 3 when 1≤n≤12
Number of people
()
Order in which people are gone
The winning number
1
1
1
2
1,2
2
3
3,1,2
2
4
3, ... Show more content on Helpwriting.net ...
Applying the equation to Josephus problem, to find we have to find as . We know how to solve as
there are only 2 people and starts meaning he has to skip and therefore dies first and wins. Now we
know that , we can see from the table above the answer is correct however we have to do it once
more to find .
We can always consider position and we have to repeat the equation until getting to the required
number. This solution can be very frustrating as you must do the same thing over and over, however
it can be programmed very easily and save a lot of time.
This method is the third and the last method to solve the Josephus problem, it is the most useful

method as it works for all bases however it is the most complicated one. In order to compare
between this method and the other two methods, we first have to write the equation in base

Place Value
The ability to make sense of numbers is necessary in order to solve a wide range of mathematical
problems, both in further education and every day life. A teachers understanding of place value and
the base–ten system is paramount for developing a student's ability to confidently approach number
operations. To underpin and guide a student's knowledge of place value, clear insight into what a
student knows about the concept, is pertinent to the continuation of learning and building of
understanding.
Rationale
Educators must draw upon their professional judgement to make astute teaching and learning
decisions, to extend a student's learning of place value and the base–ten system. Effective teachers
will; implement well–planned mathematical activities to help engage learning, use observation and
focused question to see where support is needed, and provide opportunities for growth and
understanding of place value (Department Of Education Western Australia [DOEWA], 2013). One–
on–one interviews are an invaluable diagnostic assessment strategy that unravels, not only the ...
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The essence of understanding place value, is not only knowing the values of the numbers depending
on their position, but being able to rename their values in order to problem solve (NSW
Government, 2015). For example, the number 345 could be written as 34 tens and 5 ones, 345 ones
or 3450 tenths (National Council for Curriculum and Assessment [2015], 2015). The most
significant aspect is being able to comprehend that whilst there may be different ways to write the
number, it still has the same value. The Australian Curriculum recognises this importance for more
complex learning in; algebra, fractions, decimals and multiplicative thinking (Commonwealth Of
Australian, 2009). Without this knowledge, problem solving becomes limited, resulting in confusion
and

Lesson Reflection Paper
At the beginning of the Kindergarten class the class did their normal routine of calendar time. The
teacher tells them to go to the carpet for calendar time and that gets them on task. They count the
number of days they have been in school out–loud. They also count the days with straws and put
them in a ones, tens, or hundreds pocket. The students also keep record of the number of days they
have been in school with change. They count out change to the number of days they have been in
school. They say how much each coin is worth and who is on it. Lastly, they add a gumball to their
jar of gumballs and say how many they have in total. This is a normal everyday morning routine that
involves math. TMPs 3 and 6 are used for this routine and SMPs 4, 6, and 8 are used during this
time. This activity takes around thirty minutes everyday. I think doing this everyday has really
benefited the students. A student found a dime outside and he brought it to me. He knew it was a
dime, how much it was worth, and even the president that is on the dime SMP 6. I observed a forty–
five minute second grade lesson using the Invagations program. The lesson was called how many
stickers how much money (cents). The lesson began with the students coming to the carpet and
reviewing the value of coins. After that the students were shown stickers in a base ten blocks format.
They were told to come up with an answer of the total number of sticks and write it on their white
boards SMP 7. Then they were

The Numbers And Counting System In Ancient Egypt
Numbers and counting systems are used daily by everyone. Numerous civilizations throughout
history like Egypt, Babylonian, Maya and Africa developed a unique number system. They were
commonly used to communicate numbers in an everyday life. Since then counting has changed. A
few of the earlier systems had principles that survived and helped in some way, shape or form to
create our current Hindu Arabic numbering system. It has become widespread. Back in 3000 B.C,
ancient Egypt was using the number system to calculate areas of land, distribute money, and much
more. The Egyptian counting system consisted of hieroglyphs and pictorial signs. The graphic
numbers can be of a person, animal, or plant. The hieroglyphic numbers were a written form of the
number system. It used a decimal base approach. For all the powers of ten, there was a unique
symbol. For example, the number ten was represented as a upside down "U", for the number a
hundred thousand it was tadpole, for a million it was represented by a man kneeling with his arms
raised. When reading or writing the number, it is written from right to left. It was written on
temples, vases, and stone monuments. The Egyptians used hieroglyphic numbers to advance in
architectural achievements. Math was used to make historical creations like tombs and pyramids.
Greenwald and Thomley agree that "the ancient Egyptians were also aware of fractions, which were
primarily written as unit fractions of the form 1⁄n, such as 1⁄2or 1⁄4..."

Conformity In Emerson's Self-Reliance
In the dictionary, conformity is defined as "compliance with standards, rules, or laws.". The reason I
believe that conformity is a common element in both Emerson's Self–Reliance and Gladwell's
Outliers is because in Self–Reliance, Emerson believed that to be self–reliant, you must avoid
conformity. Gladwell agrees with this in chapter 8 of Outliers, where he states that we tend to
conform to rules based off of where we are from and how we were raised. This chapter was a
perfect example because of how all those children were raised. As stated previously, a big key factor
in Self–Reliance was the concept that Emerson truly didn't want others to follow conformity. He
wanted people to remain true to themselves due to the fact that it follows along with the essence of
the Universal Spirit. "To believe your own thought, to believe that what is true for you in your
private heart is true for all men,– that is genius" (Emerson, 19). After Emerson discusses conformity,
he delves into the discussion of innocence and infants. He discusses this as a nod to nonconformity.
"Infancy conforms to nobody: all conform to it, so that the babe commonly makes four or five out of
the adults who prattle and play to it" (Emerson, 20). In a whole, Emerson's Self–Reliance was more
or less

Disadvantages Of Multiplication In Mathematics
Based on the Council of Teacher of Mathematics,(1993) "Multiplication involves the counting of
units of a size other than one." The repeated addition definition while is a useful link between
multiplication and addition is limiting if it is students' only concept of multiplication.
The meaning of the multiplication sign, "×", depends on the language of the speaker. In Japanese it
always means "multiplied by." "3 × 4" and "3 times 4" mean, "Three multiplied by four," or four
groups of three items. In English, however, the sign means either "times" or "multiplied by,"
where"3 times 4," denotes three groups of four items, and "three multiplied by four" means four
groups of three items. Therefore, in English the sign "×" has two interpretations ... Show more
Basic knowledge on numbers and Mathematical skills.
4. Pedagogical readiness is the "Students" understanding of the materials they use as they learn
Mathematics.
5. Maturation readiness is the "Students" level of mental maturity (each person passes through four
stages of mental maturity).
The use of games and concrete materials can aid in pupils' recognition of the importance of
Mathematics and its many different real applications.
Educators views on the use of concrete materials and Manipulatives in the classroom
Manipulative materials are objects that pupils can feel, touch, handle and move. The National
Council of Teacher of Mathematics( 1993), proposes that pupils and teachers often view the use of
manipulatives"as play time", but stress that using manipulatives is any excellent way to help
learners make faster connection between mathematical ideas. It is also established that learning is
enhanced when pupils are exposed to concepts in varying manipulative context. Learners who are at
the concrete level deal with manipulative materials as they discover solution to problems. Many if
not all games, involve object manipulation at some

Math Reflection
Part 1– Effective Teaching of Mathematics After taking this course I have personally learned so
much about mathematics and effective ways on how to teach certain topics in my future classroom. I
believe that this course was taught in a way for me to advocate for how I teach myself mathematics.
When I knew I was taking this class I was very nervous at first. Mathematics is not my strongest
subject so automatically I felt as though this class was going to be a long struggle for me. Once I
had a couple class periods I grew more and more comfortable with the math we were learning. I
learned to teach math to myself again. I had to teach myself math concepts that I learned way back
in elementary school that I completely forgot how to do. In this class we learned a lot about how
elementary math is done now a days. Also how now a days there is more than just using the
traditional algorithm to solve equations.
I have learned math along with my students at my placement. At my placement I observed my
students using the basic operations with distributive property, partial quotients with division and
much more. One of the really cool things I learned from my placement students is the box method
which is done with multiplication. I at first thought it would be like lattice but it's not. For example
say if you had the numbers 56 and 6, you would draw a rectangle box and split it in the middle.
Then you would write 50 in the first column on top and then 6 next to it. The number that goes on
the other side is 6. So then you would multiply 50 and 6 to get 300 in the first box. After you would
do 6 times 6 with the other box which is 36. Once you have those two numbers you add them
together to get the answer. I was never taught to solve multiplication that way so that was something
with mathematics that I learned along with my students. I do believe that thoughtful listening is one
of the most effective qualities of a good teacher. I practiced this thoughtful listening when I listened
to other people participate and show their work in class. I also practiced this thoughtful listening
when I taught my unit lesson on math word problems. I took the time while teaching my lessons to
listen to every student explain how they got

Ancient China Research Paper
In 1,500 bc, the mathematicians of China were using numbers. Long before the people of India
invented the number zero, Chinese people were using base ten counting systems. The base ten
method they used is different than the one we use today, using ancient Chinese math, the number
465 would be written as four one hundred markers, six ten markers, and five one markers. They
were also advanced in geometry, this was proved in 600 BC, when the Tangram was invented. The
Tangram is an amazing math game, that allows you to explore geometry to huge extents. This essay
tells the fascinating story of how the game is played, how it was used, and how it became known to
the western world. China is often thought of as one of the wealthiest countries in ... Show more
In ancient times, believe it or not, they were used to tell stories. The pictures made by Tangrams can
be put in order to make a scene. You can arrange the shapes in any way, and trace the picture onto
paper, and make a new figure. The traced pictures can look just like comics. You actually can think
of Tangrams as the comic books of the ancient world. Quite truly, you can make any story, from
superheroes to mathematics. This is actually quite important, creating a story using geometric
shapes can actually improve your skills in geometry. That probably explains why they were so
popular in China. "Scholars and artists thrown together are often annoyed at the puzzle of where
they differ. Both work from knowledge; but I suspect they differ most importantly in the way their
knowledge is come by. Scholars get theirs with conscientious thoroughness along projected lines of
logic; poets theirs cavalierly and as it happens in and out of books. They stick to nothing
deliberately, but let what will stick to them like burrs where they walk in the fields." –Robert Frost.
This quote highlights the fact that both scholars and artists take an interest in Tangrams. "Tangrams
are a great thing to incorporate into the mathematics classroom because they are fun, interesting,
and

The Learning Of Mathematics And Mathematics
Overview In 21st century classrooms, an educators teaching practice is vital in developing student's
mathematical knowledge. A constructivist approach is required to allow students to use their prior
knowledge to make sense of new information through hands–on activities. To effectively equip
students with the necessary skills to see connections in mathematic concepts, Big Ideas must be
employed by educators. The Australian Curriculum supports the use of Big Ideas to deepen students
understanding of mathematical content. Students learn effectively when they can see connections
between concepts. The most effective way to link an array of mathematical concepts is through Big
Ideas. A Big Idea is defined as a "statement of an idea that is central to the learning of mathematics,
one that links numerous mathematical understandings into a coherent whole" (Charles 2005). Big
Ideas are broken down into several elements. These are the name of the Big Idea, the idea central to
the learning of mathematics and the links to several mathematical understandings. Students need to
learn the mathematical understandings to allow them to understand the Big Idea (Charles, 2005).
Big Ideas are important as they should be the basis for student learning, the teaching pedagogy for
educators and encompass the Mathematics curriculum. Hiebert et al. (as cited in Charles, 2005)
agrees and explains that individuals understand concepts if they can see a link to other things they
know, which is the

Mathematics and Theology Blossoming Together
Mathematics and theology have blossomed together throughout history with many great
mathematicians also being great theologians. However, in the modern scientific era, mathematics
has become by and large secularized in mainstream academia. Although the secularization of
mathematics seems to ignore mathematics' metaphysical value, in truth, this secularization allows
for mathematics to act as a universal tool and allows the individual to attach his or her own personal
truths without marginalizing the beliefs of others, especially in education. In addition, attaching
one's personal beliefs to systems of math and logic may lead to contradicting interpretations of the
material when taken into the larger context of society, such as with the concept of infinity, the
meaning of truth and proof, or even the source of mathematics itself. In essence, the secularization
of mathematics is a necessity in our modern dynamic world and in order for mathematics to
maintain its effectiveness as a universal tool our personal beliefs must not be attached to
mathematics as a whole.
To begin with, a proper definition of secularization is necessary in order to establish the correct
connotation for the term. As denoted by the world renowned scholar in the study of the sociology of
religion, Jose Casanova, secularization is spoken of in three senses:
a.) Secularization as the decline of religious beliefs and practices in modern societies, often
postulated as a universal, human, developmental

2123 Base 3
Adding numbers in base 3 is quite simple. First of all you identify the problem, for this occasion the
problem used will be 2123 +1223 knowing that we will be adding in base three means we are
restricted to only using the numbers 0,1,2. Now we are going to start by adding 2 and 2 together
which makes four, but you can't write down four so we must now subtract 3 so that it can give us a
number to work with, and we are also subtracting 3 because it's the base we are working with. Then,
you must carry the 1 and write down a 1 down at the bottom too. Secondly, we are going to now add
the middle column, which is 1 and 2 also don't forget to add the 1 that we carried earlier.Four would
be the answer but we can't write down 4 so as previously done before we have to subtract 3 and
carry a 1 and also write down 1 at the bottom of the problem because that is the answer. Finally we
move to the last column which is 2 plus 1 and the other 1 we previously carried. The answer to this
is four again so you just repeat the step we took earlier. Since we carried the one from the left over
four we must now bring it down to our answer. At the end you should've ended up with the number
1,111 if ... Show more content on Helpwriting.net ...
First of all we have to find out what number stands for which letter in the alphabet. For this example
we are going to be using A=01 and B=001 so for example C would have to equal 0001 and D would
be 00001 and so on. Since we now have our rules for what numbers mean which letters we can
move on to writing secret messages. With our given code we can now decipher any code, so then
what would 0010100001 mean? It is easier to break down the code first. since all of our numbers
end in 1 we can start there and example of this would be the following: 001,01,00001. Now that we
have our code separated the final step is to look for what the numbers stand for in this case 01=A,
001=B and 00001=D so this must mean that BAD is our secret

Earliest Civilization is the Region of Mesopotamia Because...
A civilization is recognized as such by its form of written language. For this reason, the earliest
civilization is recognized in the region of Mesopotamia with their language of Cuneiform. This
ancient form of written language was inscribed on clay tablets that still remain in tact and are being
salvaged hundreds of thousands of years later. Even more impressive than just writing the language,
however, is the ancient Babylonians' early mathematical discoveries. These were also recorded with
cuneiform and recorded on clay tablets, and like the language, served as an early interpretation of
mathematical principles that influence arithmetic all over the world today. Dating back to the second
and third milennia BC, Babylonians were so ... Show more content on Helpwriting.net ...
The Babylonians used pre–calculated tables to assist with arithmetic. Perhaps the most amazing
aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two
tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the
numbers up to 59 and cubes of the numbers up to 32. Most frequently Babylonians utilized tables of
squares and cubes to simplify multiplication. The concept of reciprocals was also first introduced by
the Babylonians. Because they did not have a method for long division, they were able to recognize
that using their sexiagesimal system of numbers, numbers with two, three, and five, had finite
factors of which tables have been found. For numbers not containing one of the finite factors, the
Babylonians used approximation reciprocals. The pre–calculated tables method is also how the
Babylonians incorporated algebra in their number system. They were the first people to use the
quadratic equation, though not in its exact form. They used the form x2+bx=c which, when solved,
can be interpreted as x=–b/2–√(b/2)+c which more closely resembles the modern quadratic
equation. Using their arithmetic tables of squares, the Babylonians were able to interpret them in
reverse to find square roots. Because everything was a real problem, they always used the positive
root when solving. Most commonly squares were used for finding

The Impact Of Arabic Numerals In Medieval Europe
Impact of Arabic Numerals on Medieval Europe Medieval European society was changed by the
introduction of the Arabic numerals into their society. The Islamic Golden Age introduced lots of
innovative thought into the world, and eventually those ideas made their way into Europe, one of
which was the Arabic numerals. They revolutionized the way that daily tasks, like merchant
bookkeeping, and academia were approached. Medieval Europe was transformed by the Islamic
Golden Age and that is highlighted through the transformation Arabic numerals had on society.
Preceding the Islamic Golden Age, Indian culture had a revolution of thought which was seen in the
Islamic Empire. One thing from Indian culture that transcended into Islamic culture was the concept
of zero. This was something that was not considered in earlier mathematic studies. It read in "Math
Roots: Zero: A Special Case," "the Arabs recognized the value of the Hindu system, adapted the
numerals and computation, and spread the ideas in their travels." The Arabic people saw the power
in this numbering system because there was a place holder number. This concept was accepted into
Islamic thought; however, it was not received well in Europe. For the greater part of the European
society, it was a strange system, in comparison to the Roman numeral system, and was not widely
accepted. At the beginning of Arabic numeral introduction into European society, scholars and
mathematicians were primarily the only ones who accepted

Why Do Lincoln Use Significant Figures
Any good chemist would understand that when Abraham Lincoln said "four score and seven years
ago" he was actually using significant figures. The Declaration of Independence was signed on July
4th while Lincoln delivered this speech on November 19th. This means that it actually occurred four
score, seven years, four months, and 12 days before. Admittedly that does not roll off the tongue as
smoothly, so it's understandable that Lincoln would only use two significant figures. Significant
figures are the reason that the beginning lines of the Gettysburg address will forever be remembered
by Eighth graders across the country. Without the use of significant figures, numbers in chemistry
wouldn't have any meaning. I could say 2g when I'm really ... Show more content on
Helpwriting.net ...
If you are given a measurement of 4g and a measurement of 4.0001g, you would understand that the
equipment used to measure 4.0001g is much more precise than the equipment used to measure 4g.
Although significant figures tell us about the precision of the measurement, we really cannot attest
to the accuracy of the measurement unless we are sure of the correct answer. The same rule about
the estimated digit is applied when taking measurement. The way to report an analog measurement
is to write down all the digits you are "confident" in and estimate one digit further. For example, if
using a 10ml graduated cylinder with nine markings in between the numbers 1ml and 2ml and the
liquid falls between 1.5ml and 1.6ml, but closer to 1.5ml, the digits you are confident in are 1 and 5.
Therefore you estimate another digit which you think it could be such as 1.53ml. When making a
digital measurement, the rule is to always write down every digit while understanding that the last
digit is an estimation. There are also a few important differences between the course policy and the
textbook. One is that the book states that if you place a $#$#0$#$# to the right of trailing zeroes, it
would make those zeroes significant e.g. 500. has 3 significant figures while 500 has 1. This is not
true and they should be ashamed. Another difference is in the convention of marking significant
figures. In this course, Professor Halpin will underline all significant figures to identify them. It just
makes it easier to get the correct answer while rounding to the correct number of significant figures.
The book does not do

Why Is A Manipulative?
A manipulative is often used in many ways to teach mathematics such as basic addition, fractions,
decimals, order of operations. To name a few manipulatives; blocks, cards, number tiles, counting
tubes, etc...A manipulative can be taught either concrete (hands–on) or virtual. Hands–on
manipulative models are physical objects such as base–ten blocks, deck of cards, Dice games, and
Algebra tiles. A virtual manipulative is a technology that models the existing manipulatives such as
base ten blocks, rulers, fractions bars and algebra tiles to name a few. These manipulatives are in the
form of Java or Flash applets, a web base technology. Normal playing cards have so many uses in
teaching mathematical lessons. When teaching your students ... Show more content on
Helpwriting.net ...
A disadvantage of using dice in a classroom can be a bit noisy with the rolling of the dice & students
possibly dropping them on the floor. Although Dice games can be used to teach multiple math
concepts such as learning multiplication, addition and subtraction and decimals to name a few the
games can tend to get boring. The students may either get board or just play with the rolling of the
dice for fun. Base Ten Blocks has been the most used manipulative numerical system in just about
every classroom all over the world. They help elementary students visualize the base ten system
while working on basic or a more advance math lesson. One advantage of using Base Ten Blocks;
when introduced early in elementary education it helps students learn to develop their counting
skills at a quicker pace and provides a solid strong foundation for advanced mathematics. Another
advantage of using Base Ten blocks; they are great to use in guided math groups. Students are able
to develop problem solving skills while working together in groups. A disadvantage of using Base
Ten blocks; if there is just teacher demonstration and no hands on learning the students will not
grasp the concept so readily. Another disadvantage; to multiply is that any product over a certain
size cannot be represented (2002, M Freitag) Algebra math tiles help students of all learning styles
to better understand mathematical ideas. One advantage of using Algebra tiles; they are

Place Value And Decision-Making Model
Introduction This report is about place value and how it should be approached based on the
interview style diagnostic observations of year six student 'Ariel' and her knowledge and
understanding of place value, including her ability to correctly use it with mathematical operations.
The aim was to diagnose areas that required learning and offered improvement opportunities. The
aim was to then support this with the development of a teaching plan. This was achieved by
instructional activities through a series of tutoring sessions that worked towards refining her areas of
knowledge and improvement, and then extend on those ideas. After completing the diagnostic test,
the areas Ariel struggled most with were multiplicative thinking, partitioning, ... Show more content
on Helpwriting.net ...
They are selected to support and enhance the lesson objective (Reys et al., 2012, p. 43). There are
ungrouped materials that can be formed in to groups, for example, single beads or straws or
anything that can be bundled into groups of say,10, or another amount. There are also pre–grouped
materials that are already bundled to represent an amount. The use of 'manipulative' materials can be
useful visual aids and assist children grasp concepts with a hands on approach, making it easier for a
teacher to observe what a child is doing rather than thinking (Marshall & Swan,

The Binary Number System In The Hindu And Boolean Algebra
In the modern world, binary numbers have great importance in computer coding as well as in
Boolean algebra Without the application of binary numbers, computers would not be able to exist or
communicate with each other. Binary numbers include a base two number system rather than the
base ten system which is used in a math class such as algebra. A base two system means that the
only numbers that are used in that system and in binary those numbers are 1 and 0. A base ten
number system beans that numbers 1, 2, 3, 4, 5, 6, 7, 8,9, and 0 are used. All numbers can be
converted from decimal, the base ten set of numbers, to binary and from binary to decimal allowing
computers to communicate. "The Hindu–Arabic system is based on ten different symbols and is
considered to be a base 10 system. Numeral systems with different bases have found use in
applications where a different base provides certain advantages" (Lande 514). The Hindu–Arabic
system is the decimal system that is used during a math class and the binary number system is used
for different functions in computers or Boolean algebra. Due to the Hindu–Arabic system and the
binary number system have different numbering bases, they must be converted to the other. The
binary number system is a base two system which has to be converted to a base ten if it was to be
expressed in decimal, the Hindu–Arabic system. The number 10011011 in binary would be 155 in
decimal. This process would be done by writing the binary number then

Observation Is The Major Role Of The Teacher
Observation: Observation is the major role of the teacher. The teacher observes for routines and
procedure to be observed. The teacher observes for ground rule in the classroom and in individual
materials. The ground rules are opportunity for the child to develop his/her will. The teacher
observes for pattern of order in the behavior of the child. The teacher observes for respect child to
child, child to material, child to adult interaction.
Control of Error: is the built in aspect of the material also sometime found as coding on the back of
the material.
For Example, in spindle box the extra or not enough spindle left is control of error. The material like
matching numeral to number rods, teacher is the control of error. Prepare of the environment: A
prepared environment is an important role of Montessori classroom .the goal of prepared
environment is to make children safe and comfortable. A calm and orderly space enables the
children to work on various activities of their choice at a pace they are comfortable with. They also
experience a mix of freedom and self–discipline in space that is designed to meet their development
needs.
A teacher observes the child working with the materials and make changes to the environment. In
mathematic materials, extensions and additional problems are introduced to deepen child's
understanding of materials. For example, variations introduced when child is ready with basic
concepts of a particular material to support child needs.
Relation

Essay About My Math Teaching Experience
My math teaching experience went with students who are in third grade. In my group I had four
students, where was one ELL student. They were struggling in math as per their teacher, especially
in subtracting big numbers with zeroes. I had to reteach subtraction, three or two digit number, as
needed, from three digit number. And concentrate on subtractions from zeroes. Everything started
great, the kids were exiting, as of they are going out of the room. We sat down in the hall and it was
not comfortable.The kids were sitting near the wall, I was sitting in front of them in the middle of
the walking area. As of kids were sitting on the floor and me too, we were several times distracted
by other kids or adults, who was walking through the hall, it was always disturbing us. I started my
lesson with math solving problem and my students easy were able to figure out what operation that
needs to do. They all said subtraction. I said great. Then, I asked them to solve this problem
independently. Student 1, was just sitting, Student 2 used her fingers to count, Student 3 did very
fast, but not correctly and Student 4 just copied that incorrect answer. Then, I asked to share their
answers. Student 3 raised her hand and said, it was easy, we have 0–9, will be 9. Then 0–5, will be
5. Then 4–2 will be 2. So, the answer is 259. From this answer I understood, that my students
understand that ones must be under one, and tens under tens, hundred under hundred, but students

Essay On Ello
Step 14: Draw place–value blocks
Tell students, another way we can solve this is drawing place–value blocks. Tel students a line will
represent a tens rod and a unit cube will be represented by an X. (Draw and label on ELMO for
students to see)
Model how to represent 39 with place–value blocks.
Draw the three long lines on ELMO to represent the three equal rows that Nelly wants to use to
divide her 39 stickers.
Inside each line start to draw one vertical line to represent a tens rod, then draw 3 X's per line to
represent 3 unit cubes.
Each "row" should have one line (tens rod) and 3 X's (three unit cubes).
Ask students how many place–value blocks do we have per row? (13)
"You can draw place–value blocks to help you solve division problems whenever you don't have
place–value blocks to help you solve a problem. ... Show more content on Helpwriting.net ...
Step 16: Write an equation
"Now that we have modeled division with place–value blocks and drawings lets come up with an
equation."
Write on ELMO:
Step 17: Introduce mnemonic: "Does McDonalds Sell Cheeseburgers?"
Explain to students the mnemonic "Does McDonalds Sell Cheese Burgers?" will help them
remember the steps for division
Write down these steps next to the division equation on the ELMO and go over each one with
students o D– Divide (÷) o M– Multiply (x) o S– Subtract (–) o C– Check () o B– Bring down ()
Provide handout to students for them to glue in their math

Thinking, Fractions And Decimals
Teaching students effectively in areas of multiplicative thinking, fractions and decimals requires
teachers to have a true understanding of the concepts and best ways to develop students
understanding. It is also vital that teachers understand the importance of conceptual understanding
and the success this often provides for many students opposed to just being taught the procedures
(Reys et al., ch. 12.1). It will be further looked at the important factors to remember when
developing a solid conceptual understanding and connection to multiplicative thinking, fractions and
decimals.
When teaching mathematical concepts it is important to look at the big ideas that will follow in
order to prevent misconceptions and slower transformation ... Show more content on
Helpwriting.net ...
C., 2015, p. 3).
Symbolic representation using base–ten and expanded algorithms is a way to show students the
written connection to the visual models used. The partial–products algorithm is a more detailed
step–by–step process and therefore more advisable to avoid errors in students learning to grasp the
procedure (Reys ch.11.4). This process allows students to visualise the distributive property more
easily. However, the standard multiplication algorithm is quicker and acceptable for students, if the
teacher feels they have complete understanding of the steps in the partial–products algorithm.
Multiplication by ten gives students opportunity to explore larger numbers, and can also be extended
on(Reys et al. ch. 11.4). In addition, multiples of 10 give students the knowledge that all digits move
left one place and an additional place hundreths. This concept can be used to introduce the decimal
place which is also moving place each time something is multiplied by tens. Exposing students to a
range of examples which displays patterns that occur when multiplying by tens and hundreths will
generate meaning of digits moving place (Reys et al., ch. 11.4).
Visual models known as arrays or grids can be introduced early to assist students thinking by
providing a visual representation when going from adding to multiplying. In addition, arrays are a
great

Comparison Between Roman And Modern Era
From calculating digital computations to setting the foundation of modern applications, the number
system is more relevant in life and culture than most people realize. For example, people use the
number system to organize sections and chapters from literature, keep financial affairs orderly and
neatly, and compute numerous types of everyday costs. Also, it dictates how modern applications
function and how programming languages work seamlessly. Despite these incredible
accomplishments, the world would lack all knowledge of it were it not for ancient number systems.
These systems include three of the most famous ones in the world: Roman numerals, Mayan
numerals, and Egyptian numerals. Through innovation, technology, and ingenuity, ancient ... Show
more content on Helpwriting.net ...
For example: IV = 4, IX = 9, XXXIV = 34, CIX = 109, CD = 400, and MCMIV = 1904. The key to
handling this exception is to subtract the digit ahead of the power from the numeral itself [4].
Now that the basic numbers and rules have been covered, the arithmetic for addition, subtraction,
multiplication, and division can be understood. To calculate addition, one must first rewrite the
numbers to arrange them in descending order after putting them together. Next, one must learn to
ignore the subtractive system and write out individual numbers when appropriate. For example: 4 =
IIII, 9 = VIIII, 34 = XXXIIII; but 3 still equals III, 6 still equals VI, and 37 still equals XXXVII.
Third, evaluating in ascending order, one must rewrite trailing "I" digits into a simplified version.
Finally, any further required simplification is performed. To demonstrate this, consider two numbers
in addition, 13 + 66. The numbers are written side–by–side (XIII + LXVI), then put together in
descending order (LXXVIIII). The four trailing "I" digits are rewritten as IV, and this number is
rewritten as LXXVIV. The last three digits, the "VIV," is equivalent to "IX (9)," so the number is
rewritten one final time as LXXIX (79).
For subtraction, one must borrow numbers by marking out common symbols in the second number.
For example, LXVIII – XII (68–12) =

Justifications : Warm Up : Shared Book Essay
Justifications: Warm–up: Shared Book Description: Justification: The warm up for this lesson is
reading a book called, "Earth Day – hooray!" I chose to incorporate reading as a math warm–up
because it is an effective way to explore mathematical concepts, which are both engaging and
informative for the students. Integrating elements of literature into the lesson supports problem
solving and allows students to be fully immersed in the story, solving themselves the mathematical
issues that the characters fall into. Ministry of Education, 2016, endorses this idea stating that
applying visual elements of picture books aid in illustrating mathematical concepts, which are
difficult to teach with other resources, or challenging for students to comprehend. This particular
book introduces properties of place value, discussing the idea that 10 tens is equal to 1 one hundred,
which all groups need assisting with. Picture books contextualize mathematical ideas in meaningful
ways for students, while creating stories, which are relatable and spark curiosity and enthusiasm
(McGrath, 2014). McGrath further explains that learners will respond positively when stories are
not written with the intention of teaching children mathematics and will be more receptive to
learning the math message, if the thinking is second nature. As the cappuccinos are learning about
place value this warm–up is a great launching point for furthering exploring this concept and
prompts ideas for discussion and

Numbers In Norse Mythology
Numbers have had a fascinating presence throughout history. There is evidence of the use of
numbers dating back at least twenty–thousand years, such as the discovery of the Ishango bone. The
Chinese, Mayans, Romans, Babylonians, and Greeks all had numbering systems.They were all
unique and they have all played a role in the use of Numbers in their everyday lives. Numbers have
also played a significant role in magic and divination.The I–Ching system from the Chinese, the
tarot, and numerology are all some very popular forms. A Greek philosopher named Pythagoras is
credited for modern day numerology. Arithmancy is another form of divination based on assigning
numbers to a word or phrase.This can be seen in the Hebrew Kabbalah.
Numbers are all ... Show more content on Helpwriting.net ...
Odin, hung himself in Yggdrasil( the tree of life), he spent nine days and nine nights hanging in the
tree. He made this sacrifice of himself to gain power over the runes and learn of all their secrets.
During Ragnarok Thor fights the sea serpent Jormungand, after killing the serpent, Thor takes his
final nine steps and then dies from all the poison the serpent spit on him. In Norse mythology there
are nine worlds, Asgard– home of the gods called Aesir, Alfheim– the land of light elves,
Vanaheim– the home of the Vanir gods, Midgard– the human world, Jotunheim– the land where the
Giants dwelled, Svartalfheim– where the dark elves lived, Nidavellir– dwarf territory, Muspelheim–
the land of fire that Surt ruled, Niflheim– The mist world. In the center of Niflheim was the world of
the dead, ruled by Hel. Yggdrasil (the tree of life) is in the midst of these nine worlds.These are just
a few of the ways the number nine is significant in Norse mythology. The number nine is in the
Christian Bible. According to bible, number 9 has got a lot of spiritual significance. When Jesus
Christ was nailed on the cross, he dies in the ninth hour. Also Jesus appears a total of nine times to
his apostles and disciplines after his resurrection. As specified by Saint Paul, there are nine spiritual
gifts of God which are wisdom, knowledge, faith, gift of healing, to operate miracles, prophecy,
discreetness of spirits, tongues and to interpret them. Also Saint Paul listed nine fruits of spirit
which are love, joy. Peace, patience, kindness, goodness, truthfulness, gentleness and self control.
The number is is seen throughout different stories in the Christian Bible and is considered a sacred
number. In Chinese culture they like the number nine alot. The number nine to the means eternal
and everlasting. On Valentine's day a man usually gives a woman either 99 or 999 red roses to
symbolize their everlasting love. In traditional Chinese

Why Is A Manipulative?
A manipulative is often used in many ways to teach mathematics such as basic addition, fractions,
decimals, order of operations. To name a few manipulatives; blocks, cards, number tiles, counting
tubes, etc...A manipulative can be taught either concrete (hands–on) or virtual. Hands–on
manipulative models are physical objects such as base–ten blocks, deck of cards, Dice games, and
Algebra tiles. A virtual manipulative is a technology that models the existing manipulatives such as
base ten blocks, rulers, fractions bars and algebra tiles to name a few. These manipulatives are in the
form of Java or Flash applets, a web base technology. Normal playing cards have so many uses in
teaching mathematical lessons. When teaching your students how to add, subtract, sort, compute or
compare numbers, using a deck of cards gets the students working with numbers while playing a
game. There are two advantages of using a deck of cards as a hands–on manipulative: First
advantage; cards are inexpensive, can be easily put away. Second advantage; the cards can be used
as independent practice or cooperative learning. During cooperative learning the students become
involved & they are able to practice the edification adeptness such as multiplication games,
fractions, percentages and decimals. There are two disadvantages of using a deck of cards as a
hands–on manipulative; often students end up dropping things on the floor which becomes a
disadvantage because you spend time making sure

Description And Description Of A Calculator
<!DOCTYPE html> CALCULATOR body{ margin: 0px; padding:0px; } main section{margin:
20px;} #calc_body, #calc_header{ background–color: #BFBABE;} main section table {padding:
20px; border–radius: 10px; border–spacing:0px; border–collapse:collapse;} table, th, td
{display:inline–block; padding:0;float:left;} #heading{ text–align: center; font–size:23pt; color:
black; font–weight:bold; text–transform: uppercase;} main section table tr td button{ height: 70px;
display:block; width: 90px; font–size: 22pt; border:0; } .longbutton{ height: 70px; width: 180px;
margin: 0; text–transform: uppercase; font–size: 22pt;} main section table tr td button:hover{
background–color: #E990C0; font–weight: bold; } #display {width: 540px; height: 80px;
background–color: #B0EEDB; font–size: 25pt; border:none; outline:none; text–align:right; margin–
bottom:15px;} #expr{ width: 540px; height: 80px; background–color: #B0EEDB; font–size: 13pt;
border:none; outline:none; } #Trig{font–size: 26pt;} .orangebuttons{ background–color:#F99B58; }
.orangebuttons:hover{ background–color:#FBB888;} .operators{ background–color:#7B8EF8;}
.operators:hover{ background–color:#B6C0FA;} #equalsign{ background–color:#343436;
color:white;} #equalsign:hover{ background–color:#959699;} Calculator log2 Rads

The Development Of Place Value
1.3 The Development of Place Value
Place value is essential to developing number sense and without it students would not be able to
give meaning to numbers. Place value underpins important mathematical concepts, such as part–
part–whole knowledge, estimations, mental strategies, flexible partitioning, and knowledge of
multi–digit operations (Dawson, 2013; Hurst & Hurrell 2014). Frequent hands–on counting
experiences with concrete materials, models, resources and activities are mandatory to progress
students understanding of place value (Appendix A). Ross (2002, p. 420) states when students are
provided with conceptual problem–solving activities rather than procedural activities, a greater
understanding of place value is developed. Moreover, multiple embodiment experiences allow
students to work flexibly with numbers. Research shows students' ability to work flexibly with
numbers in different contexts are limited if they are not provided with "...perceptually different
models" (Reys et al., 2012, pg. 28). Hence, students' development of place value relies heavily on
conceptual learning with explicit multi–embodiment experiences and tasks using concrete resources.
However, before students can develop base–ten number and place value systems, educators must
introduce pre–number concepts including classifications, patterns, conservation, comparisons and
one to one correspondence, group recognition (subitising), and counting strategies (Rey et al.,
2012). These concepts are

Math Observation
In the beginning of my observations the students in the classroom had been practicing their addition
and subtraction in a one digit numbers. However, as the year went on my cooperating teacher started
to get her students to be solving two–digit addition and subtractions problems. Before she started
with two–digit she wanted to be certain that her students felt confident along with concrete
knowledge and examples in their one digit equations because if not, they would fall apart for two
digit equations. My teacher had opened with two–digit addition, by developing knowledge on base
ten blocks for approximately two weeks, then progressed her students to the standard algorithm. My
classroom is set up for their numeracy portion of the class that ... Show more content on
Helpwriting.net ...
My student at the beginning of two–digit addition struggled with being organized in his base ten
block drawings and would get the answer wrong because he would circle 11 ones instead of 10.
When he got the opportunity to work with the actual base ten blocks he had a strong concept of
regrouping. Before the lesson my cooperating teacher taught them how to do the standard algorithm,
which my case study student took to excellent. During the process of the standard algorithm I asked
him to walk me through what he was doing and the process. He explained that he carried the one
over because it was making a new ten and then counted twenty plus thirty plus ten instead of 1 plus
two plus three, which I felt like the was understanding the place value. I thought during the lesson if
my student was advancing ways to challenge him on this assignment would be to have him add
three two digit numbers or have him add one two digit and one three digit number. Then if I felt like
my student was starting to struggle I could have his partner help him solve the process, bring out the
base ten blocks, or have him just add one two digit and one digit numbers until he got the concept

Additional Mathematics Chapter 1 Essay
Chapter One From chapter 1, I feel a very important contribution to the development of
mathematics is the Egyptian "doubling" method. The Egyptians came up with this dyadic method
when they realized that any integer can be written as a sum of "doubling numbers" or powers of two
without repeating any of them. By doubling a number enough to add up to the value of another, the
Egyptians came up with an approach to multiplication that we should purpose today. It is also an
early introduction to the distribution property that will be used in later mathematics such as Algebra.
I find this method of "doubling numbers" as a path to multiplying large numbers that avoids the
mistakes that can be made with multiplication in our current method such as "dropping the zeros for
the next place value"; and I will surely teach this method to my middle school students as they
progress to mastering mutiplication.
Chapter Two The "Babylonian" formula from chapter 2 is what I feel is a very important
contribution to the development of mathematics. They Babylonians came up with the formula and
they did not even know it. ... Show more content on Helpwriting.net ...
Interesting that we actually get introduced to this theorem in our text earlier in time in the
Babylonian Era of chapter two. Plimpton 322 actually contained pairs of numbers in which the
square root of the difference of the pairs' squares was a whole number; what we know as today as
Pythagorean Triples. But it was Pythagoras who took this property beyond just a concept and
applied it to right triangles; where the sum of the two squares of the two shorter sides of a right
triangle equals the square of the hypotenuse. There is no way that I will teach mathematics and not
come across teaching any of my students the Pythagorean Theorem; whether it be Pythagorean
Triples, right triangles, or Trigonometry, the Pythagorean Theorem will be

The Linguists: Boarding Schools
The Linguists Linguists study language because they are concerned about the amount of languages
disappearing. In the video it states, "One language is lost per two weeks." When a language dies,
some kind of unique world could be lost. Gregory and David are visiting places that are most in
need. These places tend to be smaller in population. Younger children are typically the ones to stop
speaking the indigenous language. I believe since they're language is not as popular and most these
languages don't have a written form, it would be easier to learn a different language for learning
purposes. Maybe schools are the ones who don't teach these indigenous languages to them.
Boarding schools in some of these places bring in groups of minorities of tribes to a central location
to teach them. In the video, the boarding school in India brought in 60 tribes around the location to
teach them English, and the practice of Hindu religion which presents subtle pressure on the
children's identity. The learn many valuables skills at boarding schools but bringing in of many
different types of groups affects their learning about their origin tribe or group. When the kids go
home they do not speak their origin language as much or even at all. Older generations tend to be
the only fluent speakers of these dying languages. Parents even in their fifties or sixties are
considered to not speak their origin language because this way of education started around their
time. There is a way to resolve this conflict, and that is if every indigenous group had a school of
their own. This would be a solution but many of these places can probably not afford a school for
their indigenous group. There is a solution but it is a question if the solution could be realistic for
the communities or not. Since most of the tribes are not Hindu yet they're learning Hindu at the
school. This may show up in census of religion in areas not being of their origin religion but as
Hindu. Younger children's may mostly convert to Hindu because that is what they are taught in
school thus making their indigenous group's language one step closer to being extinct. Elicitation is
a technique of getting any of a number of data collection. This relates to the film

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