Published 2011 Revised 2019

*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 | 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 | Electric circuits | |

Differential equations | Simple mathematical models | Models of the atom |

Technology | Fitting curves to data |

Solution: We designed stemNRICH around the content areas needed for university STEM courses.

**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.

Solution: Our stemNRICH problems target these critical mathematical process skills.

Summary

To make the most of and enjoy a university STEM course students need to have a solid base of appropriate **mathematical content** coupled with an equivalently strong set of **mathematical process skills** allowing them to apply their knowledge successfully. Insufficient levels in either area
will cause students trouble.

NRICH problems differ from many standard textbook questions or interventions because there is always a focus on a mathematical process; stemNRICH takes this well-developed NRICH philosophy and applies it to scientific contexts.

It is hoped that by supplementing standard, traditional preparations with material from stemNRICH, students will arrive at university well equipped for a happy, productive and successful time.