MA6400 Education

Essay Part A

Introduction

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