MA6400 my theories. Lastly, I will briefly talk

MA6400 Education
Essay Part A

Introduction

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In this
essay I will demonstrate my knowledge and understanding of the key concepts on
Indices, I will also show the development of this topic throughout the national
curriculum (KS3-KS5) and how it is taught to pupils and the different teaching
methods used, I will also show which teaching methods are more effective and
use my research and studies to support my theories. Lastly, I will briefly talk
about the affective dimension of learning mathematics and how equal
opportunities between gender, class and culture are linked in.

Definition of
indices

Indices
are also commonly known as exponents for us to understand indices we will need a
basic explanation on what an exponent is, an exponent or index is a simple way
of expressing large numbers. For example, any number multiplied by itself is
called a square number , the raised 2 is the index and the and the 3 is
the base number or another example we can look at is  mathematicians
before us saw this as a problem and to simplify it they created indices, we
would write this equation as  in its
simplex form in other words it is shorthand, we use many symbols in maths to
simplify, denoting other meanings which allows us as mathematicians to write
formulas in a more concise and comprehensible way. These will be the first examples
students are introduced to in schools.  https://sciencing.com/history-exponents-5134780.html

Students that
are introduced to indices and exponents will be initially taught squaring and
cubing, the notion of squaring and cubing numbers go as far back to the Babylonian
times. These were apparently found recorded on a tablet
where they used symbols from their own numbering system, Sumerian, to denote
mathematical formulae. https://dcdexponents.weebly.com/history-of-exponents.html

The concept of
exponents/indices and the history of it?

Exponents,
meaning place, came from the Latin word expo. Exponents had many meanings,
however the first use of exponents ever recorded in 1544 and was written by a Mathematician,
Michael Stifel, called “Arithemetica
Integra”. In this book he was only working with the base of 2 and exploring
its powers leading to the discovery of geometric sequences. https://sciencing.com/history-exponents-5134780.html

The man that
discovered the concept of exponents was a man names Euclid, which was used in
his geometric equations and not algebraic equations, we see in the future that
Archimedes generalises the concept of powers. Time had passed
and mathematicians in the Islamic golden age had discovered algebra utilizing
the powers of two and three. Nicholas Chuquet was the first to use
exponential notation in 1484.  https://dcdexponents.weebly.com/history-of-exponents.html

 

What the Earliest Exponents Looked Like

 

 

The exponents we learn
about today in modern mathematics were discovered by a series of scientists and
mathematicians over the period of 8 centuries where each
scientist/mathematician added to the meaning and the basic concepts of the exponential
function we know today. https://dcdexponents.weebly.com/history-of-exponents.html

Nicolas Chuquet (1445-1488) invented the radical
symbol that we use to represent roots. He was also the first to represent positive
and negative powers using a raised number. Not long after Robert Recorde
(1512-1558) introduced the terms squared and cubed that are currently taught under
the National Curriculum. Later on down the line a collaboration between John
Napier and Henry Briggs created a series of logarithms to represent all numbers
as a product of a base and an exponent. E.g. log (2)=100 as 10 to the power of
2 is 100. Rene Descartes a philosopher introduced the use of superscript to
represent exponents. Sir Isaac Newton, one of the worlds most influential
Scientist, was the first to use fractional exponents and negative exponents in
1676. https://www.sutori.com/story/history-of-exponents

Leonhard Euler gave us the basic scaffolding
for natural logarithms as the inverse function for natural exponential functions
used in A-levels. In 1748 his theory for exponents suggests that the exponent
itself is a variable and quantities of this kind cannot be algebraic functions,
if they were this would suggest that exponents must be a constant. https://en.wikipedia.org/wiki/Exponentiation

How indices are
used in the real world

Indices
are used in a variety of ways, we use indices in everyday life without even
realising. Exponents are used in computer game physics, pH and Richter measuring
scales, science, engineering, economics, finance, accounting and many other disciplines.
There are several uses of exponents in real life, as well as their impact on
our understanding of the modern world around us.

·        
Indices in Computers: power of 2
exponents is the basis of all computing which is done in “Binary” or base two
numbers, see below.

 

 

·        
Compound Interest: exponents are widely used in investing and
finance. Money invested that earns interest on interest follows an exponential
rate of growth to produce large amounts of money. E.g retirement funds and long-term
investments

 

·        
Richer scales: ricther scales are used to measure how powerful earthquakes
are.  The actual energy from each quake is a power of 10, but on the scale
we simply take the index value of 1, 2, 3, 4, etc rather than the full exponent
quantity.

 

·        
Science: indices are also used a lot in science, Very large numbers,
like the distance between planets, or the population of countries, are
expressed using powers of 10 in a format called Scientific Notation”. E.g. the distance
between stars and planets, earth to moon distance is about .
 

http://passyworldofmathematics.com/exponents-in-the-real-world/

Ks3 –
introduction to indices, how it is taught and different teaching methods with
supporting research

Key stage
3 will be where pupils will be initially presented indices/exponents. Teachers will
have many obstacles to tackle when they are teaching this topic. There are many
ways a teacher can show a student how to calculate indices and why we use
indices.

In the national
curriculum it states that students will need to combine their numerical and mathematical
capability from key stage 2 and extend their understanding of the number system
and place values to include decimals, fractions, powers and roots. It will initially
be difficult for students to understand the concept of indices therefore be
able to select and use appropriate calculation strategies to solve increasingly
complex problems. Also in the syllabus students will need to use conventional notation
for the priority of operations, including brackets, powers, roots and
reciprocals; use integer powers and associated real
roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish
between exact representations of roots and their decimal approximations; interpret
and compare numbers in standard form  where n is a positive
or negative integer or zero.

 

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf

 

Ks4 – how
indices become more complicated, how is it taught and different teaching
methods with supporting documents

Key stage
4 is where it becomes more complex for pupils to calculate indices

KS4

consolidate their numerical and
mathematical capability from key stage 3 and extend their understanding of the
number system to include powers, roots {and fractional indices}

calculate with roots, and with integer {and
fractional} indices

calculate exactly with fractions, {surds}
and multiples of ?; {simplify surd expressions involving squares for example
12 4 3 4 3 2 3 = ×= × = × and rationalise denominators}

calculate with numbers in standard form A
10n , where 1 ? A < 10 and n is an integer • simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by: § factorising quadratic expressions of the form 2 x bx c + + 2 ax bx c + + , including the difference of two squares; {factorising quadratic expressions of the form } § simplifying expressions involving sums, products and powers, including the laws of indices {estimate powers and roots of any given positive number} select and use appropriate calculation strategies to solve increasingly complex problems, including exact calculations involving multiples of ? {and surds}, use of standard form and application and interpretation of limits of accuracy https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/331882/KS4_maths_PoS_FINAL_170714.pdf   Show working examples in each stage Talk about  issues of equality of opportunity and inclusion with respect to gender, class and culture. Conclude essay why it is effective to teach the topic in the curriculum and how it benefits pupils.   Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history's most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf       http://www.resourceaholic.com/2014/12/indices.html    

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