There are three kinds of mathematics: the math that’s taught, the math that’s learned, and the math that’s needed in the 21st century STEM workplace. With support from the Advanced Technological Education Program at the National Science Foundation, Michael Hacker, Co-Director of the Center for STEM Research at Hofstra University, and I organized a conference to study why those three “maths” are not the same.
Held in Baltimore from January 12th through the 15th, the conference attracted 46 attendees drawn from three groups: math educators, STEM content instructors, and STEM employers. Three fields of STEM employment were represented: Information and Communication Technology, Biotechnology, and Advanced Manufacturing.
There is ample evidence (see, for example, “Still Searching: Job Vacancies and STEM Skills”) that companies in these and other STEM-related fields are finding it difficult to find qualified employees for entry-level jobs. This is due in part to the poor math skills of prospective candidates, and – perhaps even more telling – their lack of confidence in their ability to “do” math. In this context, the objective of the meeting was to solicit from employers examples of problems that prospective recruits often could not solve. The meeting would then collectively examine those problems, identify the underlying relevant mathematical concepts and skills, and explore possible explanations for why high school and even two- and four-year college graduates find the problems so challenging.
A complete reporting of the findings of the conference must await our analysis of the data we collected over the course of two days of intense discussion. However, it is already evident that real-world challenges, such as those described by the employers, differ from the math problems that most students encounter in formal school settings.
An example – one of many – may illuminate these differences.
An employee in a communications technology firm is tasked with providing a commercial space, consisting of several offices as well as other rooms, with wireless Internet access. The tools available consist mainly of access points and routers, the former connecting multiple devices using radio frequency communication, the latter directing information between those devices and an Internet service provider (ISP). Access points have a limited range and their locations must be selected so that those ranges overlap, providing connectivity to every device on the network as well as to one or more routers. Routers, in turn, require connectivity to the ISP.
At first glance, the problem seems simple enough: just place the access points close enough to one another so that their ranges overlap. But real-world complications soon arise.
To save money, the number of access points should be minimized. Further, they require power so installing them in some locations may result in wiring expenses. The range of each access point may be affected by the materials used for interior walls or by metallic structures such as elevators or vaults. Privacy and security concerns dictate that access to the network be restricted, as much as possible, to the premises of the customer. Some locations within those premises – e.g., conference rooms – may require greater bandwidth than others.
These and other real-world considerations are not, strictly speaking, mathematical in nature, but insofar as they constrain the set of acceptable solutions, they require mathematical skills – e.g., modeling – that may be foreign to many would-be network technicians. Moreover, although the calculations required consist primarily of arithmetic operations on numbers (signed integers, decimals, and fractions), the semantics behind these calculations – unit conversions, use of the Pythagorean Theorem to compute point-to-point distances, algorithms for computing overall costs – are not explicitly called out in the statement of the problem.
Thus, even though the problem appears to require no more than middle school math and Algebra 1, it differs from the problems commonly encountered in traditional classes in those subjects.
- The statement of the problem does not contain all the information required to solve it and may in fact contain irrelevant information.
- The mathematical concepts and skills required are not spelled out (in contrast to the problems found at the end of the chapter in a math textbook, all of which involve the specific concept covered in that chapter).
- The problem is multi-step and involves multiple variables.
- The problem may have many solutions of varying utility, rather than a single “right” one.
A major finding of the conference was that the kind of mathematics encountered in each of the three domains represented (ICT, biotech, and manufacturing) involved contextualized problems similar to the one described above. Thus, an important barrier to success in these fields may arise from the features of such problems that we have identified.
Are there ways in which educational technologies such as those pioneered, deployed, and investigated by the Concord Consortium could help students to acquire the relevant, contextualized problem-solving skills? A major outcome of the conference may turn out to be a number of proposals aimed at answering that question.