How do you view mathematics?

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Your view of mathematics:

Read the following six statements, and find three statements, which best describe your view of mathematics and its applications:

• Mathematics consists of applying the rules of logic to solve problems involving numbers
• Application of mathematics involves using and manipulating numbers to enable you to solve problems
• Mathematics is the study of numbers, shapes and formulae representing numbers and shapes
• Mathematics is a complex framework of ideas which enables you to make sense of the world
• Mathematics is an abstract system of thought that is based upon a process of formal reasoning
• Mathematics is concerned with solving problems about numbers and using the results to help you understand the world

How do you view mathematics feedback

Many people view mathematics as a collection of facts, whether of rules, procedures, theorems, definitions, formulae or applications. Each fact can be clearly stated. We know, for instance, that 8x7=56 or that to add two fractions together you need to start by ensuring they share a common denominator. An application of mathematics then involves using these facts to solve practical problems, such as how to determine the optimal price at which to sell some goods. Ticking the boxes for statements 2, 3 and 6 in the quiz might suggest this is how you view mathematics.

There is, however, a contrasting view that mathematics is more a system of interconnected ideas than a collection of facts. This view starts with the understanding that mathematics is underpinned by logical thought, so that we can guarantee that answers to mathematical problems are actually correct and can explain why they are. This view holds that different ideas within mathematics and its applications are inextricably linked with each other.

Now have a go at this…

Read the following six statements, then find three statements, which best describe your approach to studying mathematics and its applications:

• Mathematics involves learning lots of formulae and rules
• I try to make sense of a new application of mathematics by making sure I understand all of the underlying principles
• I learn mathematics by memorising all of the important results and by practising lots of problems
• Reading my note several times over is my preferred way of studying mathematical applications
• The important thing is to see how well you understand an area of mathematics by tackling some difficult questions
• If I am introduced to a new concept, I look to see how it relates to other mathematical ideas

Click here to see what implications how your approach to mathematics and its application may have on your studies.

Your approach feedback

One approach to studying mathematics and its applications is to try and memorise all of the relevant information and to practice carrying out the tasks you are asked to complete. Memorising that 3x1=3, 3x2=6, 3x3=9 and so on is a classic example of this approach. Statements 1, 3 and 4 in the quiz have been designed to identify this approach to study. If you picked out most or all of these three statements, then is it also true that the previous diagnostic test indicated that you tend to view mathematics and its applications as a collection of facts? Theory suggests that if you view mathematics as a collection of pieces of knowledge, then it will seem quite reasonable to simply memorise each piece in turn.

A second approach to studying mathematics and its applications is to concentrate on making sure you understand what is going on. For this approach, memorising how to solve an equation is less important than understanding why the solution is valid and how the solution relates to other mathematical ideas. So you do not just use the letter ‘x’, you also make sure you understand what it represents. This approach to study is picked out in statements 2, 5 and 6 in quiz 2. Is the case then that if you view mathematics as a coherent system of ideas then you also tend to look for the meaning of ideas?

Consequences for your study

Most students will tend to approach their study either on the basis that most mathematics is a collection of facts, which need to be memorised, or on the basis that mathematics is a system of ideas, which need to be connected to each other. The two contrasting approaches are summarised in the figure below. But which of these two contrasting approaches is more likely to be successful?

Memorise facts or procedures

Look for connections between ideas

Memorise facts or procedures or Look for connections between ideas

One of the most robust all findings from educational research is that students who just try to memorise facts achieve lower grades than those who primarily seek meaning. If you think about it, the finding is entirely unsurprising. Facts are only relevant if they actually mean something to us, otherwise we quickly forget them.

It is also the case that some aspects of mathematics and its applications are not amenable to memorisation. For instance, Newton noticed a link between an apple falling and the mathematical idea that we now refer to as a vector. There is no formula you can memorise that will enable you to understand how gravity can be modelled by the idea of a vector.

Furthermore, approaching mathematics as a collection of facts encourages you to think that you can understand each fact on its own. Unfortunately for this approach, most mathematical ideas are neither quite so simple nor so isolated. Even the idea of an equation involves several mathematical ideas. In practice, you only appreciate most ideas when their links with other ideas apparent. This leads us to two of the most fundamental of all intellectual skills: the skills of analysis (taking something apart) and synthesis (joining facts together).

How you approach the study of mathematics and its applications will make a big difference to how well you succeed. Concentrating on memorising facts and procedures is not a recipe of success because it does not take account of the nature of mathematics and its applications.

Effective study of mathematics is characterised by looking for connections between ideas rather than just memorising facts and procedures.

This content has been written by Peter Kahn, author of Studying Mathematics and its Applications.