David M. Bressoud November, 2009
This past September, the Association of American Medical Colleges (AAMC) and the Howard Hughes Medical Institute (HHMI) issued new guidelines for the scientific preparation of pre-medical as well as medical students, AAMC-HHMI Scientific Foundations for Future Physicians . They have moved away from the traditional medical school recommendations that focus on the titles of courses, instead explaining the competencies that students need if they are to succeed in medical school and as physicians.These two organizations have a very powerful voice, and pre-med requirements greatly influence the curricula of biology departments, many of whose majors arrive with the expectation of preparing for medical school. Mathematics departments need to pay attention to these recommendations.
Engineers and Biologists
Before discussing the recommendations for the mathematical training of pre-med students, I want to emphasize a point I have made before: The mathematics curriculum of the first-two years as it is now constructed was designed to serve the needs of future engineers. It also does a pretty good job of meeting the needs of students heading into the mathematical or physical sciences, but, at least on a national level, it is engineering programs that fill these classes. Service to biology departments is often an afterthought. Even many of the calculus classes that are labeled as being for biologists are really just traditional courses with biological examples tacked on.
While we cannot afford to ignore the needs of engineering departments, our mathematics departments—if they are to thrive—must also pay attention to the needs of the biology departments. This can be seen very dramatically in the following graphs showing the pattern of intended and actual majors since 1980. The first graph is based on national surveys conducted during fall freshman orientation. The second graph is based on bachelor's degrees awarded in the year ending in the spring of the designated year.
Source: The American Freshman 
Source: National Center for Education Statistics 
The messages of these graphs are (1) that engineering and the biological sciences are responsible for far larger numbers of students than the physical, computer, and mathematical sciences combined, and (2) that the dominant role of engineering among the STEM (science, technology, engineering, and mathematics) disciplines is being challenged by the biological sciences.
The AAMC-HHMI Recommendations
In their recently released report, AAMC and HHMI begin with eleven "overarching principles" that give an indication of where they are going to place their emphasis as they discuss the preparation of pre-med students. These principles include the need to understand uncertainties and risks and the ability to examine the medical literature critically, which includes the use of statistical inference and experimental design.
For the expectations of student competencies at the time of entering medical school, the first competency describes quantitative reasoning and mathematics.
Competency E1: Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world.
This competency is comprised of seven learning objectives.Two of them deal with basic quantitative literacy (#1) and logical and algorithmic thinking (#6), neither of which is commonly addressed by mathematics departments for pre-med students. Four of the objectives deal with statistics (#2–4 and #5 in so far as one considers statistical models). Only two of these objectives are related to calculus (#5 and #6), and these objectives focus entirely on modeling dynamical systems. Finding derivatives and integrals appear nowhere. Knowing how to read a differential equation and to understand a model of a dynamical system are what these students need. This is completely in line with the recommendations of the biologists back in 2000 when they met for the MAA's CRAFTY Curriculum Foundations Project in Biology [4, pages 15–18]. What biologists need most from the mathematical sciences is statistical knowledge, and fairly sophisticated statistical knowledge that includes multivariate analysis. From calculus, they need both far more and far less than what is taught in the standard engineering syllabus.
Many colleges and universities are working on these courses designed specifically for the needs of biological science majors, Macalester College among them. Such courses should not supplant what we do for engineers. Nor should they form a separate and independent stream. Students who start in a calculus for biologists course must be able to use it as their first course in mainstream calculus. It is possible to construct a mainstream calculus sequence that begins by building an understanding of dynamical systems. The US Military Academy at West Point has been doing this for two decades. Many of the Calculus Reform projects (see [5, 6]) developed successful materials around this approach.
The learning objectives and examples for Competency E1 as given in the AAMC-HHMI report are listed below:
1. Demonstrate quantitative numeracy and facility with the language of mathematics.
- Express and analyze natural phenomena in quantitative terms that include an understanding of the natural prevalence of logarithmic/exponential relationships (e.g., rates of change, pH).
- Explain dimensional differences using numerical relationships, such as ratios and proportions.
- Use dimensional analysis and unit conversions to compare results expressed in different systems of units.
- Utilize the Internet to find relevant information, synthesize it, and make inferences from the data gathered.
2. Interpret data sets and communicate those interpretations using visual and other appropriate tools.
- Create and interpret appropriate graphical representations of data, such as a frequency histogram, from discrete data.
- Identify functional relationships from visually represented data, such as a direct or inverse relationship between two variables.
- Use spatial reasoning to interpret multidimensional numerical and visual data (e.g., protein structure or geographic information
3. Make statistical inferences from data sets
- Calculate and explain central tendencies and measures of dispersion.
- Evaluate hypotheses using appropriate statistical tests.
- Evaluate risks and benefits using probabilistic reasoning.
- Describe and infer relationships between variables using visual or analytical tools (e.g., scatter plots, linear regression, network diagrams, maps).
- Differentiate anomalous data points from normal statistical scatter.
4. Extract relevant information from large data sets.
- Execute simple queries to search databases (e.g., queries in literature databases).
- Compare data sets using informatics tools (e.g., BLAST analysis of
nucleotide or amino acid sequence).
- Analyze a public-use data set (e.g., census data, NHANES, BRFSS).
5. Make inferences about natural phenomena using mathematical models
- Describe the basic characteristics of models (e.g., multiplicative vs. additive).
- Predict short- and long-term growth of populations (e.g., bacteria in culture).
- Distinguish the role of indeterminacy in natural phenomena and the impact of stochastic factors (e.g., radioactive decay) from the role of deterministic processes.
6. Apply algorithmic approaches and principles of logic (including the distinction between cause/effect and association) to problem solving.
- Define a scientific hypothesis and design an experimental approach to test its validity.
- Utilize tools and methods for making decisions that take into account multiple factors and their uncertainties (i.e., a decision tree).
- Critically evaluate whether conclusions from a scientific study are warranted.
- Distinguish correlation from causality.
7. Quantify and interpret changes in dynamical systems.
- Describe population growth using the language of exponents and of differential calculus.
- Explain homeostasis in terms of positive or negative feedback.
- Calculate return on investment under varying interest rates by utilizing appropriate mathematical tools.
 HHMI-AAMC. 2009. Scientific Foundations for Future Physicians: Report of the AAMC-HHMI Committee. Association of American Medical Colleges. Washington, DC. www.hhmi.org/grants/sffp.html
 Pryor, J. H., S. Hrtado, V. B. Saenz, J. L. Santos, and W. S. Korn. 2007. The American Freshman: Forty Year Trends. Los Angeles: Higher Education Research Institute, UCLA, supplemented by the 2007 and 2008 reports. www.gseis.ucla.edu/heri/40yrtrends.php
 The data are from the National Center for Education Statistics. 1990–2009. Digest of Education Statistics. US Department of Education. nces.ed.gov/programs/digest/
 Ganter, S. and W. Barker. 2004. Curriculum Foundations Project: Voices of the Partner Disciplines. MAA:Washington, DC. www.maa.org/cupm/crafty/cf_project.html
 Smith, D.A. and L. C. Moore. 2009. Calculus: Modeling and Application. 2nd edition. Online only: www.math.duke.edu/education/calculustext/
 Callahan, J., D.A. Cox, K. R. Hoffman, D. O'Shea, H. Pollatsek, and L. Senechal. 1995. Calculus in Context. W. H. Freeman: New York. www.math.smith.edu/Local/cicintro/cicintro.html
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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and President of the MAA. You can reach him at email@example.com. This column does not reflect an official position of the MAA.