What is mathematics?
Mathematics is a social science that has many central logical and natural scientific disciplines. By the most general definition, mathematics includes “the study of quantity, structure, space, and change”.
Examples of the logic branches include formal logic, mathematics of language, philosophical logic, and mathematical reasoning.
The natural sciences include mathematics applied to geometry and algebra, applied mathematics (applied in domains such as engineering), probability theory, statistics; pure mathematics (purely abstract or not for a particular physical application).
Mathematics is central to all sciences, engineering and pure mathematics.
There is a broad mathematical curriculum which has no beginning point or established end point; it evolves over time as human knowledge expands.
While pre-college education in mathematics is often viewed more as training than a formal academic discipline in itself (e.g., how to use calculators), undergraduate programs generally include algebraic structures and abstract algebra, analysis including calculus, linear algebra and differential equations; geometry and trigonometry; analytic number theory and real analysis (see also Measure); set theory and combinatorics.
Further subspecialties exist outside of academia as applied mathematics and mathematical modeling.
Importance of mathematics
Mathematics is one of the most important subjects to study. The more mathematical understanding a person has, the more they can learn and do in life. Nevertheless, mathematics is often thought to be complicated and tedious for school kids.
The benefits that come from studying math are numerous and worthwhile:
- Mathematicians have helped build technology like satellites, public transportation systems, storm prediction models, video games, the list goes on and on.
- A mathematician’s mathematical intuition can guide complex decision-making processes in businesses or governments when faced with challenging problems (How many oranges will we need?). Such decisions might otherwise have less reliable outcomes as only gut instinct would be used to establish a numerical answer without an underlying understanding of applicable numbers or their relationships to each other.
- If a person wants to “think smarter,” studying mathematical principles like the number systems, angles and shapes can be very enlightening on their own in ways that will influence all areas of life, such as reading or interacting with friends.
Mathematics is often thought to be one of those subjects necessary to grind through only to pass a test and move on rather than try to enjoy it for itself. The purpose of this article is twofold:
- To explore some basic but essential mathematical topics that students might not have seen before or fully appreciated (a few even without having been identified directly as mathematics yet),
- To provide ideas and examples that teachers could use when teaching these same topics.
By experiencing mathematics in an enjoyable context and applying it to real-world problems, a student can be introduced to the beauty of mathematical principles in ways that are not usually emphasized as directly. As a result, students will be more willing to make learning these topics a priority rather than just passing them up to get on with other subjects.
The “Mathematics” section begins by giving examples of how a mathematician’s thinking could be applied outside the classroom – general problem solving using numbers (equations). We then focus on exploring math concepts related to numbers such as units of measurement, counting, addition/subtraction/multiplication/division, percentages, and proportions before closing with some ideas about geometry (shapes) — arguably all essential concepts for students which are often neglected in everyday classroom learning.
The “Math” section uses words in its title instead of mathematics to point out that there are various ways to explore math concepts that are as important as memorizing a formula or solving an equation – studying the way numbers work (why does 7 x 8 = 56?) is just as worthwhile, if not more so, than knowing that something is true but never having understood why it works the way it does, etc.
What follows is some material on topics often thought of being outside the realm of mathematical interest yet all intimately related and surprisingly interesting when seen from their own perspective.
General Mathematics Topics
Mathematics covers a vast range of subjects that can be engaged with different levels of complexity. There are numbers and counting on the one extreme, which can be studied in depth using operations like multiplication/division or even logarithms.
On the other hand, a person can look at the number system as it is used in daily life to make decisions:
- How many people?
- How much money do we have?
- What time will I need to leave my house to reach my appointment on time?
In between these two extremes of math topics lie ideas such as understanding proportions – how accurately something is represented by less than 100% (only 80% of children did well on their exam). Each of these levels of complexity has its own set of concepts that are important for students to understand.
An example of general math topics is this concept of looking at things in different ways – the spatial view of a shape is just as crucial as its numerical properties for understanding what it represents.
Mathematics Topics within Numbers
Numbers are a fundamental part of our daily lives, and their impact goes far beyond simply telling us how many things exist or how much money is owed to us. Numbers are used in all conceivable human endeavors – finance and commerce (banking), engineering (calculating the amount of materials needed for a building project), social sciences (how accurate are your results from a survey of 100 people?), etc.
Each number has its own mathematical properties that make it distinct from other numbers such as dividing by zero, taking 1/3 + 1/2 = 2/3 vs. 1 = 0.33333…, or having negative exponents, etc. using percentages such as 60% instead of just saying 30 out of 60 items were correct is another excellent way to get students thinking about math concepts that involve working with numbers rather than merely memorizing facts.
Table 2 is an example of how math topics can be organized around numbers in order to understand the unique properties and behavior of each one of them: Natural numbers (1, 2, 3…), Integers (more complex than counting on your fingers – often negative or larger than counting on your fingers…), Rationals (like decimals are fractions made up by a numerator and denominator) and finally Irrational numbers such as π.
Mathematics Topics within Geometry
As stated earlier, geometry is usually taught toward the end of elementary or starting high school when students study shapes like circles and rectangles. On the other hand, looking at mathematics from this angle gives more time to explore concepts such as symmetry, proportionality, and area relationships between figures. This is a good example of how mathematics ideas are interrelated and can be taught at various levels according to each person’s needs and interests.
Mathematics topics within Geometry also include surfaces which makes sense since many shapes (like a cube) rely on them to exist (a rectangle has two surfaces while a circle is flat with zero). Thus, for example, studying the graphs of quadratic equations using surface models versus simply looking at a table that shows the outcomes of each number on an x and y-axis.
Mathematics Topics within Probability
Probability is a branch of mathematics that deals with reasoning about randomness and uncertainty. This covers a lot of everyday situations such as winning a lottery, dealing with the stock market, or even quantum physics, where subatomic particles behave in predictable ways (sadly). The knowledge of how to deal with uncertainty and be comfortable making inferences based on incomplete information is very useful for students who will face many situations in their lives requiring them to make decisions.
Table 4 uses probability as another lens through which math concepts can be viewed – complex numbers are often described as being real numbers plus imaginary one. In fact, if you add a real and imaginary part to the complex number formula, this indeed represents a probability – just plug in anything as a value for x and y.
While math is traditionally categorized as an independent subject taught within the school, these are only one way of looking at it. It can be viewed from many perspectives – Numbers can be used to describe Geometry (or its opposite) or how probability helps us make decisions about everyday life. Having such a wide range of topics to choose from allows teachers to customize their lessons depending on the students’ individual needs and interests, thus making them more memorable. Students will also learn that there are many ways to solve problems and not just rely on memorizing facts alone with no context or meaning behind them.
A big problem with education today is the memorization of facts and procedures for solving problems. While it is very important to be able to do these things well, this does not mean that students should feel comfortable only performing tasks repetitively as if they were robots rather than human beings who can think critically and creatively. In a field where technology changes rapidly, we need thinkers who can adapt new information and put it into context to solve new problems while maintaining an active consciousness about how society will benefit from their work.
Math topics such as arithmetic, geometry, and algebra are traditionally taught within a school at specific grade levels despite research showing that many children find numbers exciting by age 9 or younger. It also shows that topics like probability, logic, statistics and finance are better taught after the age of 10 as part of a social studies curriculum. This makes sense since financial literacy is best learned when kids have reached puberty because their brains are capable of learning complex abstract ideas in the context of real life rather than simply problems for which there is a single right answer. For example, children can learn about different types of investments (or gambles if you will) and explore how they feel about risk versus rewards despite what adults may say to do.
Branches of mathematics
Mathematics can be segmented into different branches. Each of these branches has a wide range of topics that they focus on.
The branches of mathematics include:
- Algebra: Algebra is a branch of mathematics that allows us to solve problems using letters instead of numbers.
- Arithmetic: Arithmetic is a branch of mathematics that focuses on arithmetic operations such as addition, subtraction, multiplication and division.
- Calculus: Calculus is a branch of mathematics that includes the study of limits, derivatives and integrals. It also allows us to describe how things change over time – whether it be velocity or acceleration when moving an object through space or temperature changes when heating up something like water in a pot on a stove. Calculus can be further broken down into 3 main sub-topics which are known as differential calculus, integral calculus and multivariate calculus (which combines both types). In fact, calculus was first discovered by Archimedes and is considered to be one of the most important areas of math.
- Number theory: Number theory is a branch of mathematics that focuses on numbers, including a wide range of topics such as counting, prime numbers, divisibility, and even how many different shapes there are for number sets (e.g. triangular numbers). This may seem boring, but it actually applies to real life – For example, to solve crimes, we use digital fingerprints (which are made up using only 0s and 1s) to identify criminals or determine their whereabouts based on DNA evidence left behind at the scene.
- Geometry: Geometry is a branch of mathematics that focuses on the study of shapes and their properties.
- Probability: Probability is a branch of mathematics that allows us to use math to predict the likelihood of something happening – rather than just assuming it will happen. For example, if you toss a coin in the air what is the probability that it will land on heads? Unfortunately, many people are afraid or even opposed to using statistics or numbers when making decisions about their lives because they do not understand how these things work. They feel like statistics can be “fudged” (or altered) for someone else to get the result they want, but this couldn’t be further from reality as statistical analysis must take into account multiple different variables that can then be used to determine the likelihood of something happening. For example, in the event that someone is accused of a crime, police collect evidence to determine whether they are guilty or not – They can then use statistical analysis (or number theory) to determine if other people have done the same thing in the past – allowing them to compare the new data with historical data and determine if there is enough similarity between the two groups in order to make an assumption about what happened.This branch of mathematics has been growing ever since its discovery because it allows us to understand social issues like poverty and homelessness better and improve our understanding of human nature by revealing patterns or trends. This is especially useful in areas such as environmental science when determining how many resources are left on earth prior to running out – as well as in the category of public health, where it can be used to determine if someone has a high risk of developing heart disease, lung cancer or even diabetes.
- Cryptography: Cryptography is a branch of mathematics that focuses on secure communication over an insecure medium (such as the internet) using advanced math techniques such as polyalphabetic substitution ciphers created during World War II to keep important information safe from being intercepted by others.
Cryptography also plays a big role with military applications because it prevents enemies from intercepting secret transmissions and spying on undercover operatives trying to bring down terrorist organizations like Al Qaeda or ISIS. However, nowadays, cryptography has gone well beyond just military applications and is used in all aspects of modern life – From online banking to video streaming services such as Netflix and even checking your email.
This is another area of mathematics that has grown exponentially over the past century because it provides us with a greater sense of security (knowing our information can’t be decrypted by others) while also protecting our privacy from being exposed to people we do not wish to see it.
Cryptography is also closely related to cryptanalysis, which studies methods for constructing and breaking codes and ciphers.
Note: Cryptography is mostly about securing communication, but some types of math cryptography focus on keeping digital data secure as well – using very advanced mathematical algorithms.
Now you know what is mathematics and have a brief understanding on how it works.