The Mathematical Problems Faced by Advanced STEM Students

Mathematics is critical to the study of any STEM subject; indeed, historically the development of science, technology, engineering and mathematics has often gone hand in hand.

The scientist or engineer needs to embrace mathematics in order to get the most from their studies. Unfortunately, students often struggle with the mathematical aspects of their scientific degree courses. In this article we explore some of the main mathematical problems arising. Far from simply a lack of content knowledge, we believe that the main area of concern is in mathematical process skills.

Problem: Students don't know enough maths!

Whilst preparing stemNRICH it was clear that sometimes certain content knowledge was lacking: those teaching biology, chemistry, physics and engineering courses often claimed that students didn't know enough about various topics in mathematics. Sometimes this lack of content knowledge was obvious: students in engineering need to know about complex numbers; other times it was graded or more subtle: biologists needed to know more about graphs and equations. Whilst these various topics obviously varied across universities and courses, interestingly, there was a surprising large overlap between the mathematical needs.

The following core topics seemed to emerge across many disciplines:

Topic	Easier application	Harder application
Measurements	Interpreting an image	Units
Estimation	Real world contexts	Problems with missing data
Powers	Orders of magnitude	Half lives
Equations and graphs	Growth curves	Scientific curves
Areas and Volumes	Approximating natural shapes	Packing structures
Proportional reasoning	Working out a dilution	Gas laws
Logarithms	Working out a pH	Buffers
Geometry	Packing problems	Spherical triangles
Fractions and decimals	Genetics	Error bounds
Data and statistics	Pattern spotting	Confidence intervals
Probability	Combinatorics in chemistry	Scale invariance
Calculus	Finding maxima and minima	Rates of change
Matrices	Transformations	Crystal symmetry structure
Vectors	Statics	Bond angles
Complex numbers	Problems with quadratic equations	Electric circuits
Differential equations	Simple mathematical models	Models of the atom
Technology	Fitting curves to data	Monte-Carlo methods

Problem: Students can't apply their knowledge!

Beneath any issues which might arise in knowledge of content, many students with good grades in mathematics seem to find it difficult to apply the mathematical knowledge that they might have. Why would this be the case?

It seems that there are several main reasons, common to all disciplines:

• 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 in scientific mathematics 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 meaning to real-world meaning
  Students who are very skilled at mathematics might have trouble seeing how to relate the mathematical process to a real-world context; this hampers the use of common sense, so valuable in quantitative science.
  
  
• Lack of ability to make approximations or estimations
  Real scientific contexts are rarely simple. In order to apply mathematics predictively, approximations or estimations will need to be made. To make approximations or estimations requires the student to really understand the meaning and structure of the mathematics, along with the underlying scientific meaning.
  
  
• Lack of multi-step problem solving skills
  Scientific mathematics problems are not usually clearly 'signposted' from a mathematical point of view. The student must assess the physical 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 practice
  There are two ways in which lack of practice can impact mathematical activity in the sciences. First is a lack of skill at basic numerical or symbolic 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 a scientific context.
  
  
• 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 underperform.
  
  
• Lack of mathematical interest
  Students are hopefully strongly driven by their interest in science. If mathematics is studied in an environment independent of this then mathematics often never finds meaning and remains abstract, dull and difficult.