by Peter March
We are interested in systemic change in postsecondary mathematics education. Over the last several decades there has been a remarkably large - and documentable - amount of local change in mathematics departments across the country. Yet, local change has not led to global, systemic change.
The status quo is stable. Postsecondary mathematics education takes place in four main institutional types: elite private universities, large public research universities, four year colleges and comprehensive universities; and community colleges. In each institutional type, to a greater or lesser extent, the following two structural effects make systemic change difficult.
First, mathematics curricula in the first and second years of undergraduate instruction are rigid objects that are difficult to change in other than superficial ways. There are two main reasons for this. First, any given course, say freshman calculus, has a lot of stakeholders (engineering, physical sciences, life sciences, social sciences, and business) all of whom have multiple needs and demands that, taken together, create an over-determined curriculum. Second, stakeholders want the mathematics curriculum to be delivered in efficient, timely fashion so as to minimize time to degree for STEM majors. This leads to mathematics courses that are dense, linearly ordered exercises in delayed gratification: courses stripped of engaging detail, hands-on student involvement, and connections to other disciplines.
Second, mathematics departments are managed as net revenue centers rather than net cost centers. The cost of instruction and the cost of doing research in mathematics are low and the volume of instruction is high compared to other disciplines, especially the laboratory sciences. Colleges, that is to say college deans, use surplus net revenues generated by mathematics instruction to offset net operating deficits in the laboratory sciences where the cost of research and the cost of instruction are high and the books don’t balance at the department level. Deans fear, not without reason, that significant change in the way the mathematics department does business will negatively affect the college’s bottom line.
The net result of these two effects for a typical mathematics department is quite discouraging. For example, curriculum reform that adds something new is often demanded by one stakeholder but opposed by another stakeholder if some topic they consider essential is removed to make way. Similarly, pedagogical innovation is often encouraged by the dean, in principle; but discouraged, in practice, if the cost of instruction increases.
Transformation is possible. Despite the discouraging nature of the status quo, there has been a lot of innovation in post-secondary mathematics instruction over the last several decades. Much of it has been stimulated by self-studies of the national mathematical sciences community, or prosecuted in ad hoc fashion where inspiring local leadership has emerged on campus, or funded more systematically on a national level by a series of NSF programs.
There is a lot of demonstrated good will here and, in my judgment, an abiding interest in the mathematical sciences community to tackle of the problem of increasing student learning, student understanding and student achievement in mathematics. Let’s take this as the axiom of innovation; namely, that the mathematics community will continue to innovate despite the discouraging status quo. (For the latest innovation on my campus, see here: http://youtu.be/28TiYQPktM0)
As described above, there are structural features at the campus level that render local, ad hoc attempts at innovation sub-critical; that is, either they do not catch or if they catch on one campus they do not spread from one campus to the next leading to systemic change. In a statistical mechanical analogy, local, ad hoc innovations are like thermal fluctuations in a potential well. But structural features on campus conspire to make the potential well so deep that the system is unlikely to reach a critical transition state and, therefore, unlikely to have a shot at passing over the potential barrier to a new equilibrium. We need to lower the potential barrier, somehow.
They only possible way forward is to bring all the stakeholders to the table at once. This has to be a multi-way conversation between mathematics departments, STEM departments and college administrations at the local level that is “lifted up” via professional societies or the National Academies or the White House to the national level. If we continue to force mathematics departments to broker what should be multi-way, campus-wide conversations as a series of bilateral one-offs, and be “talked to” for their troubles at the national level, then we’ll fail. We’ll also fail, in this time of diminished budget expectations at the federal and state levels, if there’s a significant net price tag. So, for the sake of argument, let’s formulate a premise of institutional chiropractic; namely, that by properly aligning the campus body politic we can create the conditions under which systemic change is possible at a manageable cost.
Continuing the slightly absurd analogy a little further, the notion is that institutional chiropractic ought to lower the potential barrier systemic to change sufficiently that native curricular and pedagogical innovations on campus become critical fluctuations that achieve a transition state; at nearly constant energy.
Caveat: Contradicted Expectations. Doing so will contradict stakeholders’ expectations. There is nothing wrong with that, per se; perhaps it’s even necessary. But people whose expectations have been contradicted are prone not to hear or to understand or to engage. That’s a fatal flaw in any plan that turns on communication and cooperation. Here’s a partial list of potentially wounded parties:
- Mathematics faculty; who will have to learn enough science and engineering to talk intelligently with their STEM colleagues about what their students need; be willing to teach flexibly and differently; and be willing to revise the overly rigid freshman mathematics curriculum
- STEM faculty; who will have to learn enough mathematics and statistics to talk intelligently with their mathematics colleagues about what their students need; be willing to take ownership of their piece of the problem; and to recognize that mathematics is a discipline in its own right, not just a technology in service to their science
- Students; who will have to adjust to the fact that plug-and-chug, drill-and-drop, pattern matching just won’t do; that mathematics is the language of science; and that to learn a language properly you must try to speak
- Deans; who will have to learn that you must, occasionally, feed the goose that lays the golden egg!