Mathematics Teaching Experience: Quadratic Equations and Place Value

This report provides a detailed reflection on a student's experience teaching quadratic equations and place value. The first part focuses on the student's preparation, delivery, and challenges faced during a teaching session on quadratic equations, including the initial nervousness, the process of l...Read More

Numerical Theory
Framework

PART A
Reflection on attaining session of Quadratic equations
My class tutor decide to give us assignment which related with enhancing our teaching
skills. For this purpose she divided us in groups. And give us task to take session of teaching on
topic of mathematics. For taking secession of mathematics, I decided to choose Quadratic
equations.
My session was held on Friday. Have 4 days for preparation. When teacher given us this
task at that time I felt nervous as well as excited too, because I like to represent myself in front of
students or other persona however Maths is one of the subject I always have fear that how I can
pas in this subject, due to high level of numerical and difficulties theories I am also ways afraid
of completing task or assignment of Mathematics. However but this time it is essential to
participate in each session which taken by our class tutor.
Even though maths is a subject which I always afraid however Quadratic equations is
one of the topic which I like the most while I solve mathematics numericals. I was very nervous
before this session, I decide to give 6 to 7 hours to each day for learning all the essential thing
about the Quadratic equations. For this purpose I have decide to use my books as well as with the
use of various website and tutorial I have learnt all the essential information regarding Quadratic
equations. I also learnt from various You Tube video how session is taken in front of students
regarding Quadratic equations .
On the day of given session on Quadratic equations, I was very afraid , I took suggestion
from my class tutor and from my friends regarding how to give session in front of the whole
class, even though my communication skills are good still I felt nervous just because of the
subject of my presentation. The time when I am starting to give my teaching session,
firstly I introduced my self in front of whole class, I given them short introduction regarding
what the students learn from my class session as well as all the essential point on which I discuss
during the time of my session.
Even though I felt nervous but once I started given my lecturer in front of student, all my
nervous, goes away as I have read the whole topic with small details related with Quadratic
equations, thus I felt confident during the time of representing my session. Duriring ther time I

given students all the essential information regarding what Quadratic equations is and how it is
developed, as the term Quadratic equations is a Latin word which means square and is is use to
formulate standard form of equation. Which define as ax² + bx + c = 0, in which x is consider as
unknown variable which valued is find by using this calculation and value of a is not equal to
0, if it is taken as 0 then this equation is not consider as liner equation. I also given them all their
information regarding how numericals are solve as well as issue arises during the time of solving
numericals related with Quadratic equations.
The concept is too wider, thus during the time of my session I give all the essential
information to my students regarding all the models which I learnt and read out about Quadratic
equations. Mathematics is not easy subjects as it is hard to understand and their will be many
issue arise during the time of solving queries however internet help in clearing my doubts.
However at them time of ending of my session when student ask queries and put question cross
question related to Quadratic equations, for a second I felt nervous however I given my answer
with full of confidence and solve their queries I also not able to answer few question, but my
tutor at that time help me to giving answer to students regarding Quadratic equations.
My overall experiences was good, I have learnt so many things after compensation of my
group session related with teaching. AS it useful in build confidence and enhance my
communication skills. Due to this session my relation with my co- team mates as well as with my
class tutor get strong.
This session useful in overcome my fear related with mathematics as it will help in
boosting my motivation related with reading and understand concept of mathematics. My skills
of solving numerical is get enhance, however I face many issue I am not able to give answer few
questions which student ask, as my whole concept of Quadratic equations is not clear. I fail
difficulties during the time of finding solution of equations as well as I am not able to understand
how theses queries of students solve. Which impacted on the presenting of my overall session as
my confidence goes how at that time.
Even though I have face difficulties but my overall experience was too good as it useful
in clear my concept of maths and overcome my fear, which will useful for future teaching
session as next time I have clear all my doubts and focus on increasing my communication skills
as well as enhancing my mathematical numerical skills in order trot represent effective
presentation in front of the whole class.

PART B
Place value is defined as a position of a digit or integer which tells its assigned value,
which is significant in mathematical communication being a universal language. This system is
introduced in order to develop understanding in regard to number system and their meaning in
numeral value. These place value includes places like ones, tens, hundreds, thousands, ten
thousands and so on (Reinholz and Shah, 2018). It plays significant role to identify position of a
digit given in a sequence. There are various importance of place value system in historical as
well as current mathematical learning .
Varying use of historical development of place value system in mathematical concept is
considered as a debatable technique. In traditional times, mathematicians did ample of study
varying sizes of numbers which raised development of place value system. In current time,
students are not required to make number system but need to interpret its structure to understand
its sense and operations (Nataraj and Thomas, 2009). To investigate this technique, it has been
considered use of different combinations of historical development of large numbers and place
value number system in order to enhance student's understanding and knowledge related to place
value structure.
Historical development of decimal number system and relation with other numbering systems
Decimal numeral system also known as base-ten positional numeral system which is
considered as standard concept for denoting integer as well as non- integer numbers. It may also
referred specifically to numbers after decimal separator that is if π is extended in numeral value
then it extends up to two decimal places i.e. 3.14 in mathematics (Benjamin, 2017)
There are various numeral systems of ancient times which uses ten with its powers to
symbolize numbers as following concept that there are only ten fingers on hands and therefore
people continued to count with their fingers (Szkudlarek and Brannon, 2018). Examples of such
structures can be greek numerals, hebrew numerals, roman numerals, brahmi numerals and so
on. Large and complex numbers were not easy to represent in given ancient systems and only
best mathematicians were competent to divide these numbers.
This issue was resolved with introduction of hindu- arabic numeral system for
presentation of integer as well as non integer numbers( decimal digits). Also during Indus Valley

civilization(c. 3300- 1300 BCE), standardized weights were formulated which were based up on
ratios like 1/20, 1/10, 1/2, 1, 2, 5, 10, 100 and so on. In the meantime their standardized ruler was
said to be Mohenjo- daro ruler which was segregated into ten equal parts. Also Egyptian
hieroglyphs(c. 1625- 1500 BCE) had used pure decimal based system which was also followed
by Cretan hieroglyphs(c. 1625-1500 BCE) being closely based over Egyptian model. There were
also various other techniques which were used in ancient times in regard to decimal based
system.
Other than above mentioned invention, decimal fraction was actually developed by
Chinese at the end of 4th century BCE which was later extended to middle east and till Europe.
Chinese written decimal fractions were said to be non positional (Yang and Qiao, 2019).
Decimal fractions are also called rational numbers that may be expressed in fraction whose
denominator is said to be power of ten. Decimal number system is somehow related with other
numbering systems such as binary number system, octal number system and hexadecimal
number system. Sometimes, it becomes necessary to convert decimal number into other form of
number system as per the requirement of respective situation.
Critical analysis of difficulties and misconceptions arising from place value and number systems
Problems and misconceptions being faced by various learners from concepts of place
value and number system is developing confusion in understanding concepts of both the
structure (Armand and et. al., 2019). This reflected that the concept was creating negative impact
over the learner's mindset. These primary conceptual systems provides base of learning in the
field of mathematics. Therefore it becomes necessary to overcome these misconceptions by
formulation impactful strategies in this behalf. These strategies could be written, mental as well
as visual in nature. By overcoming these barriers learners will develop optimum understanding
of the respective concept of place value and number system. These two concepts are analysed in
the given manner:
Place value system
Importance of place value system in primary segment of mathematics has been reviewed
over the years by various teaching experiences. It is regarded as a basis of complete number
system which helps to identify actual position of a digit to ascertain its numeral value. The
concept of place value is generally misunderstood by individuals either in written format,

mentally or visually (Lugosi and Uribe, 2020). There are misconceptions in regard to its better
understanding, developing actual learning criteria, usefulness in practical. These
misunderstanding are being faced by various learners while evaluating the concept of place value
system. In order to overcome this factor, various strategies have been drawn to develop better
understanding concept of the place value system (Avwokeni, 2018).
Written strategy: There are problems being faced by learners in regard to place value
system which they can overcome by implementing written strategy in various manner. This
technique will help them to develop better conceptual understanding in relation to the respective
problem. One of the written strategy could be setting up activities which will enable students to
work in regard with after/before/between numbers which will assist in understanding place value
system as a whole. Other manner of learning place value system could be identifying missing
number. It is a method where from the ten places in this system, some of the places will be blank
and they should be identified through applying mathematical general rules.
Mental strategy: In understanding place value system, generally mental strategies are
developed in mind in order to make it easy to understand. Individuals can memorize the order of
digits to overcome this issue. Also general way to look up to place value system is counting.
That means organising digits in groups of tens, hundreds, thousands, etc. to keep track of
counting a given set. It becomes way easier when student memorize this sequence in their
mindset and reflect it whenever needed.
Visual Strategy: Misconceptions in regard to place value system can be eliminated
through application of visual strategy being an appealing approach. Various visual strategies
includes hundred chart puzzles, place value spinners, jigsaw puzzles and so on. It becomes easier
to understand methodology of place value through implementation of such strategies as it
provides visual impactful presentation of figures which needs to be placed in a sequence or order
to reach an ultimate result. This way it becomes effective technique for development of overall
concept for a student.
Number System
It is a procedure to represent an integer or non integer digits in s specific manner.
Therefore, it is important to understand its concept with standardized format. It consists of
various forms of number presentation that is binary number system, octal number system,
decimal number system and hexadecimal number system. These forms of number system plays

vital role in formatting of a respective value (Eletxigerra and et. al., 2018). Their formulation are
different from each other and are of vital importance in mathematics. Being an important concept
there are various misconceptions that have developed in learner's mind. These misconceptions
can overcome through implementation of appropriate strategies in an appropriate manner.
Written strategy: It is important to understand the mathematical usage of number
system. By developing its proper base, it will become easier to implement written strategy with
the use of missing number technique and also drawing of cubes, blocks and other picture to
develop written strategy in regard with number system.
Mental strategy: This strategy is important from point of learner's capacity of grasp
things at a time. Therefore, it is important to develop mental strategies which will enhance the
level of understanding at its peak (Yang and et. al., 2018). Presence of mind plays vital role in
this concept so it is required to develop understanding of each segment of this system in order to
memorize its concepts in long term. Mental strategy includes fast calculation, computational
estimation, effective problem solving and so on. It also includes additions and subtractions as
well as doubles of decimals, also using various patterns for recognising numbers. Use of
multiplication knowledge and facts related to number system.
Visual strategy: It involves visual based strategies in order to overcome misconceptions
in regard to number system. These visual strategies could be used to understand the concept of
this system that will lead to elimination of any unwanted problems and misconceptions. These
strategies may include use of circle, triangle, square based activity to create number system. It
will ensure feasible problem solving in reference to number system by developing sense of
interest. Also with the use of abacus it will become easier to overcome the misunderstanding of
the relevant concept effectively. There are other visual tools that can be used in order to make it
interesting while ignoring the misconceptions in this regard. The use of blocks, puzzles,
electronic tools, tiles, etc. can help in developing better understanding of number system. This
way better development of the concepts can be marked by learners in order to reach optimum
solutions in reference to number system technique.
There is positive Base 10 is a place value scheme and it includes ten digits between 0 and 9. In
the decimal number, units, tens, thousands, thousands and on and on reflect consecutive
locations to a nearest integer. It is represented in decimal system by this method. The keep the

two numerical system is often referred to it as the binary digits, in which there are actually two
binary digits, i.e., 0 and 1. Explicitly, a lambda of 2 is the standard base-2. The quantities listed
in this method are referred to as binary figures, a mixture of 0 as well as 1. 110101, for instance,
is a binary integer. The basis is 8 in the alphanumeric character’s number line and to reflect
quantities it utilizes digits from 0 to 7. In software applications, octal digits are widely used. The
translation to metric of an agglomeration is much like the integer transformation. Numbers and
symbols or interpreted in the hexadecimal method using base 16. The quantities are first
described throughout the hex system, much as in the counting point, i.e. 0 to 9. Then, using the
alphabet letters of A to F, those number is calculated. The figure below summarizes the
description of quantities in such a scheme of hexadecimal digits. Numbers can be expressed in
all of the divisions of the number line, including decimal, binary, hex, etc. Every number that is
expressed in any of the forms of counting system also can conveniently be translated to another.
To know how and when to translate numbers from binary to decimal or vice versa, presynaptic to
decimal and conversely, and opcode to binary, review the extensive lesson on binary number
converts.
PART C
A critical analysis of major theories of learning and their development
Learning can be defined as procedure that bringing together both the impact and
experience of cognition, emotion and the environment in case to acquire, increase or make
modification in one's knowledge, skills, value and views. For the effective practice of numeracy
in class room people use different types of learning theories that help to understand things such
as:
Situated learning and learning mathematics: It is a learning theory which is related to
teaching. In this theory requires to presented all the detailed manner. It tends to have
characteristics of project based learning and problem based learning. It also presents to tie in
closely with basic ideas for problem solving. Therefore, in problem solving specify particular
domain independence and require to collect all the basis information.
Pros:
Cons:

Cognitive learning theory: This theory defined that how the human mind works at the
time of learning. In this theory mainly concentrate on how information is processed according to
brain and learning arise by the internal processing of information. It depends on the idea that
people mentally procedure the information they gather instead of simply responding to stimuli
from their atmosphere. It is a learning theory that helps people to learn different mathematical
problems and analysis that how external factors impact on the learning activities in direct and
indirect manner. The aim of the theory to increase earning capability and use mind for optimal
thinking, understanding and retention. Thus, it become easy to manage a lifelong habit of
continuous learning. Here are mentioned some pros and cons in detailed manner such as:
Pros: It is increasing learning abilities and enahnce life long learning in positive manner.
This theory helps in brianstorming and calculate big questions easily.
On the basis of this theory get right outcomes and assure about the different questions that given
by teacher to their students.
It is improvng problem solving skills in positive mannert and get advantage to sort out questions
easily.
Cons: Without understanding people can not easily link different information that can be
difficult for staudents to defined it in proper manner.
It claim that mind is not doing as computer sometimes it is doing wrong and get wrong results
and argues that human are different from computers.
It is often implies according to computer models and working accordingly (Zhou, Zafarani Shu
and Liu, 2019).
Social learning theory: This theory suggests that social behaviour is learned by
analysing and reviewing the behaviour of others. It is mainly used by the student in which learn
from other students and observe their teachers that how to solve different problems. Students
learn from anyone, teachers, parents, peers, siblings and even celebrities. In this theory
environment plays essential role in learing and most human behaviours is learned as per the
observation by modelling activities.
Pros: It is mainly used for students that learn from others around them and the media that can
can offer some explanations as how social norms are trasmitted.

It is beneficial in understanding a range of behaviours like how students come from different
environments and learn in same manner and their gender role by imitating role models that theu
recognise easily.
This theory provide relaistic and flexible position than is recommended through the behavioristic
approach as it identifies the role play in providing partcular framework in environment.
It helps to develop effective environment where student easily acquire konwledge as soon as
possible.
Cons: The mian reason of this theory that learn from around the people but sometime student
learn wrong things that impact on their mnd in negative way.
It is basically related with observable behaviours that can be easily measured but impact in
negatively.
Through this theory people learn wrong things and follow short cuts to calculate mathematicals
problems.
Analysis of how these theories have led to current principles of effective practice in teaching
mathematics and numeracy showing engagement with recent scholarly literature
These theories have led to current principles of effective practice in teaching mathematics
and numeracy. Such theories are related with the principles of effective practice and there are
mentioned effective practices in teaching such as:
 Set up mathematics goals to concentrate on learning.
 Apply tasks that promote reasoning and problem solving.
 Use and connect mathematical representation
 Facilitate meaningful mathematical discourse.
 Support productive struggle in learning mathematics
 Elicit use evidence of student thinking
 Pose purposeful questions

These are learning practices which are used by the teacher for the numeracy activities and
relate with the different theories that are discussed above. There are mentioned various principle
of effective practices such as:
 Variety: Without any session identical to previous one and range of approaches should be
utilised to support boost motivation and enhance skills in various manner (Kardong-
Edgren Oermann, and Rizzolo, 2019).
 Progression: The demands in every session will be gradually enhanced and assure that
skill development does not hit a plateau.
 Specific: As per the approach for effective practice in teaching required to be tailored to
fit the skill and experience levels of the performer. It should match their
 Measurable: The main long term goal for the developent is that achieve all short term
goal in every session should all the measurable. These are including figuers that mistakes
it easy to see whether goals has been reached or not.
 Achievable: According to this approach work on the performance and increase practices
of numercay activities to get hardest during session as they know progress is within their
reach. It can supports to assure about the efficiency and hardest thing at the time of
session.
 Time: For effective practicing require to give proper time in learning activities and proper
utilisation of all the resources which are required in different activities. Therefore, it
should not be too long as tiredness may increase and skill levels may drop along with
confidence.
 Recorded: When teacher conduct a class that time prepare all the notes in a diary that
helpful in fiture activities and help in good practicing. Along with it helps in progress and
present feelings after session and help chart skill development according to tracking goals
at the end of session (Alalwan and et.al, 2019).
On the basis of these principles increase practices of effective learning and these are
related to the learning theories of mathematics. When teacher, teach methematics so that time
require to use different theoroes of learning that helps to understand of different tricks and use
for future development. Along with use different types of principles that helps in learning and

effective practicing. Through these principles and theory increase learning of students. Along
with different numeric activities helps in mind exercise in proper manner and calcyulate different
things effectively. Moreover it supports in various learning procedure and get right results as per
the learning theory. Some times these theory impact in negative way but right calculating helps
in get right results effectively (Becker, 2020).
Two examples of numeracy learning activities from own experience and use learning theories in
regard of beliefs and practice of different activities
Numeracy is the capability to implement maths concepts in all areas of life that can help
people to develop their numeracy and maths skills through every day activities such as, counting,
talking about length & weight and looking at shapes. There are using different examples for
teaching and learning of numeracy learning activities such as:
Number hunt: It is a fun activity to motivate people to find out and recognise numbers
in their environment. It is good learning activity of numeracy and great way of motivation
learning at the classroom. This will help to develop skill of number recognition and keep a tally
to recognise how many of each number find on way. This activity conduct by teacher for the
improvement of numeracy skills. It is good activity of number learning and helps me in class and
increase my knowledge that numbers are hunting in different manner. After that solve different
exercise to get right answers of different questions. It is beneficial for us because it helps to
focuses on learning intention recognise numericals 1 to 5. The maths mastery challenge works
aboard
Crypt arithmetic Puzzle: In this numeracy activity, the digits are replaced by letters of
the alphabet and goal is for students to uncover the puzzle by determining the digit for every
letter. At the time of teaching and learning out class teacher conduct these types of activities that
helps to do mind exercise and increase out skills in effective manner. The rule of playing this
activity is, every letter represents a digit between 0 and 9. A letter cannot represent various digits
and a digit can not be represented by various letters. The numbers are starting with a zero and
provide one solution to the puzzle (Rogers and et.al, 2021).
For the numeracy activities use different types of learning theory because it helps to
understand how to numeracy activities conduct in specific manner. With the use of cognitive
learning theory use own mind and find out the number from different pictures that provide by

teacher to us in Number hunt activity. These learning theories are effective in these activities
because on the basis of these theories learn various tricks that use at the time of numeracy
activities. Some times it was not effective because every learning theory is not provide good
results and create problem at the time of learning. There are mentioned beliefs of teaching and
learning mathematics such as, mathematics learning should concentrate on practising procedure
and representation basic number combinations. We need only learn and use the same standard
computational algorithm and the same prescribed methods to sort out algebraic problems. This
time role of student is to learn information that is presented after that utilise it to sort out daily
problems on home work, tests and puzzles. The social learning theory student learn from each
other so it impact negatively because if any student do not get right answer in regard any
question so it impact on other students and learn wrong things (Smedley and Hoskins, 2020).

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Online

Developing understanding of number system structure from the history of mathematics[online]
available through <https://link.springer.com/article/10.1007/BF03217547>

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