# Mathematical Issues for Physicists and Engineers

Mathematics is critical to the study of physics and engineering; indeed, historically the development of mathematics, physics and engineering have often gone hand in hand.

The physicist or engineeer needs to embrace mathematics in order to get the most from their studies. Unfortunately, students often struggle with the mathematical aspects of their physics or engineering degree course: 8 key reasons for this are provided below.

stemNRICH was specially designed to address these 8 problem areas by providing meaningful, rich and interesting mathematical science problems for students to engage with prior to arrival at university.

### Background Mathematics

There is a great deal of mathematical content knowledge which a physicist or engineer needs to know. Some of the more advanced skills required are

 Mathematics Applications Equations All of physics! Calculus and differential equations Dynamics Vectors Forces, statics Complex numbers Wave equations, frequency analysis Matrices Stress and strain, 3D graphics Logarithms Sound intensity Logic Digital circuits and computing Estimation and approximation Checking answers, setting up problems Statistics and probability Statistical physics Geometry Statics, mechanics ... and problem solving skills Setting up any problem!

In addition to this strong base of advanced skills, any physicist or engineer needs to have very strong number, computation and general maths skills. Unfortunately, even a good grade in maths might not be sufficient to support the underlying physics once a student begins university. Why is this?

### Eight mathematical problem areas

Suppose that a physics or engineering student achieved a good grade in GCSE mathematics or AS mathematics. Why would such students struggle with the mathematical aspects of physics or engineering? There are several possible reasons:

• Overly Procedural thinking
• Mathematics exams can often be passed by learning the content procedurally. This means that students can answer certain types of question by following a recipe. The problems with more complex physical applications arise because even minor deviations from the precise recipe cause the student to fail to know what to do.
• Lack of ability to translate mathematical answers to physical interpretation
• Even students who are very skilled at mathematics might have trouble seeing how to relate the mathematical process to a real-world context. An important part of this is interpreting answers and realising when a mathematical description breaks down. Lack of skill in this area hampers the use of common sense, so valuable in quantitative science.
• Lack of ability to translate physical situation into the correct mathematical description
• The physical world is intrinsically mathematical and the process of modelling the world involves extracting the correct mathematical description from a physical scenario. This has always been the most tricky part of physics and engineering, and a lack of total confidence in the mathematics exacerbates the difficulties.
• Lack of ability to make estimates or approximations
• Physics and engineering contexts are often quite complicated. In order to apply mathematics predictively, approximations will often need to be made. To make approximations requires the student to really understand the meaning and structure of the mathematics.
• Lack of problem solving skills
• Physics and engineering problems are not usually clearly 'signposted' from a mathematical point of view. The physicist or engineer must assess the situation, decide how to represent it mathematically, decide what needs to be solved and then solve the problem. Students who are not well versed in solving 'multi-step' problems in mathematics are very likely to struggle with the application of their mathematical knowledge.
• Lack of experience with calculation or difficult contexts
• There are two ways in which lack of practice can impact mathematical activity in the all of the sciences, especially those rooted in physics
• First is a lack of skill at basic numerical manipulation. This leads to errors and hold-ups regardless of whether the student understands what they are trying to do.
• Second is a lack of practice at thinking mathematically in genuinely difficult scientific contexts.
• Lack of confidence
• Lack of confidence builds with uncertainty and failure, leading to more problems. Students who freeze at the sight of numbers or equations will most certainly under perform.
• Lack of mathematical interest
• Students are hopefully strongly driven by their interest in science and its real-world applications. If mathematics is studied in an environment independent of this then mathematics often never finds meaning and remains abstract, dull and difficult.
If you are a student aiming to study physics or engineering at university, do any of these areas seem like they might possible cause you some problem? If so, why not take a look at some of the stemNRICH problems and start thinking about these matters now?