July 30, 2018
Student-Centered Mathematics: Getting Buy-In from Business Faculty
In my previous post, I summarized the change we made in the math requirements for business undergraduates on our campus. We went from very typical requirements (Finite Math, Business Calculus) to an atypical requirement, a new class we built called Mathematical Analysis in Business (bearing the number MATH 1112).
Don’t let the generic name fool you. The new class is a drastic departure from the old.
In MATH 1112, students work on problems from business contexts that have an underlying mathematical structure and are informed by data: from currency markets, store operations, tax codes, financial markets, and Kickstarter campaigns.
The problem-solving work in applied settings is not tacked on as a sidebar that shows how methods they are learning might be applied. The applications are the course: here’s a business context and relevant data. Can you put the pieces together to answer the questions that naturally arise in the context?
How much leakage is there when you convert currencies? Which products are the major drivers of profitability? How much tax do you owe? How much did your investment grow? How many crowdfunding backers do you need to hit your goal?
The motivation driving the design of the new course was the question, “What kind of math do business professionals need to master?” With that framing, one might guess that the business faculty would eagerly embrace the change.
But college campuses change slowly. Due to the complexity and size of our operation, there is a built-in conservatism to all things curricular. Meaningful change is controversial and slow.
One advantage I had in making this change is that on our campus, schools and colleges can set their own requirements. Thus, I only had to change a rule applying to 3,500 undergraduates, not 25,000. I only had to convince 100 faculty from a few business disciplines, and not thousands from across humanities, science, and engineering. Still, minds are hard to change.
The main objection that I encountered was whether the change constituted a dumbing down of the requirement. That concern took different guises. Sometimes it appeared as indignation about relinquishing the calculus requirement. Sometimes it appeared as a concern over whether the class was more software training (Microsoft Excel) than critical thinking. Sometimes it appeared as a question on behalf of our top students: would they bristle that their Advanced Placement (AP) test scores would not exempt them?
I have a deep conviction that the change is not a dumbing down. The change is a move from something that is conventionally accepted as hard to subtler, but I think more important, forms of challenge. Here I explain the basis for that conviction, and how I went about persuading my colleagues that this change was a good one.
In my earlier post, I shared that I had been the longtime chair of the business school’s undergraduate curriculum committee. In that role, I had led a review of the math requirement. For that review, I used a survey where I asked our faculty to tell me how much they relied upon each of the topics in the required math classes. The results of that survey were striking: almost no one was expecting mastery of calculus topics as a pre-requisite for classes.
Those survey results were my most powerful evidence in the discussion about the role of a calculus requirement. Did students really need calculus to succeed in business curriculum and careers? My survey told me no, and those results were pretty convincing.
What is it about calculus that makes it accepted as sufficiently hard math, the universal proxy for quantitative aptitude? Calculus sits at the end of a sequence. Algebra 1, Geometry, Algebra 2, Trigonometry, Pre-Calculus, Calculus. This was the sequence at my high school in Southern California in the 1980s. This was the sequence at my sons’ high school in Colorado in the 2010s. This sequence is an entrenched hierarchy, and calculus is credibly harder because it builds on what came before.
But somewhere along the way, we let this ladder of courses substitute for our own analysis of what is most important, not just what is perceived as hardest, and I wanted to reopen that discussion in my school.
Where does our new class fit on this sequence? The mathematics in the class does not rank very high in the established sequence. There is a lot of Algebra, some Algebra 2. There are non-linear functions, mostly quadratic and piecewise linear functions, but not other conic sections or trigonometric functions. There is no differentiation or integration.
I don’t apologize for those absences, because I reject the calibration of the class on the standard sequence yardstick. The challenge in the course is integrating three skills: recognizing which mathematical skills are appropriate in a situation, performing the calculations correctly, and using Excel efficiently to find solutions. The level of the math skills is relevant for only one of those three parts, the second one. The real challenge and the real value from the course lie in students being able to do all three together. In other words, being able to attack a messy, real problem and use relevant data to answer the natural questions that arise.
Leading up to a faculty vote to approve the course, I regularly shared information about the work-in-progress on the course materials. I always showed three things in those communications, mapping to the three skills. I showed the areas of application and the types of questions (such as those at the beginning of this post, currency exchange, etc.), I showed the math skills, and I showed the Excel skills.
For example, in the module where students analyze trends in the stock price for the company Amazon (ticker symbol AMZN), here are some of the math skills in context:
- Write expressions that correctly use the order of operations.
- Understand the difference between absolute and percent gain (growth) and loss and calculate these values.
- Understand how combinations of percentage gains and losses adjust prices (e.g., up 10%, down 10%).
- Calculate, interpret, and apply ratios (e.g., Adjusted Close Price to Close Price). [These prices refer to the stock price context, i.e., they are not generic mathematical terms.]
- Apply growth rates in one quantity (e.g., Adjusted Close Price) to another quantity (e.g., initial investment value).
- Solve an equation of a single variable. Solve symbolically and confirm solution numerically and graphically.
- Calculate and interpret Compound Annual Growth Rate (CAGR).
- Perform geometric extrapolations.
This list shows the range and variety of the relevant mathematics for this module. In decomposing and listing the skills, we include even elementary skills like applying the order of operations. Most of our students won’t trip on that while they do the work. Therefore, we don’t bore that majority by reviewing or drilling on that basic skill. Those students can get right to work finding the value of investments in AMZN stock, a central question in this module. However, if students don’t have that skill mastered, it will become apparent as they work, as their models will yield incorrect answers to questions. They are given the opportunity to discover their own errors, and with support, correct them. They are given another chance at mastery of this foundational skill, hopefully in a context where they can see why it is essential to get that right.
For other skills, like applying ratios, the fundamental mathematical operations are division and multiplication or addition (i.e., basic ones), but the understanding of how to apply them in the given context is challenging. You have to be able to figure out which operation to do when. The setting is naturally complex enough that for most students, it is simply less work to gain the understanding than to develop and memorize a set of rules. In the best parts of the course material, we tipped the scale in the right direction: understanding, not memorization. It’s hard to get that right, and I don’t claim that we did it perfectly in all parts of the course. But we know what we are aiming for: giving students the opportunity to demonstrate mathematical maturity in shifting from seeing math as a set of steps to memorize to seeing math as a flexible and powerful approach to answering interesting questions.
I used examples like these to explain why the math skills we identified are both essential and appropriately challenging. In communicating with colleagues, I also was careful to separate out lists of Excel skills from lists of mathematics skills. I wanted to show that the two sets of skills are related but distinct. For that stock prices module, here are some of the Excel skills:
- Use relative and absolute cell references (proper use of the $ sign).
- Use VLOOKUP() to find a value from an array given a unique identifier. (Advanced but more flexible version for look-ups: composition of INDEX() and MATCH()).
- Use logical formulas like IF().
- Change format of displayed values (e.g., dates displayed as numbers to short date).
- Graph data and add a trend line as appropriate.
I could speak about the separation between math and Excel in the abstract, but I found that the best way to communicate the distinction was to make it concrete with examples. Such examples went a long way to convincing my colleagues that our new class had serious and important mathematical content, that it wasn’t “just” an Excel course, even if the most challenging mathematics topics didn’t adhere to the traditional hierarchy.
I am really proud of my colleagues and the leadership of our school for being willing to take this leap. Inertia supports the standard requirements. We overcame that inertia and did something that serves our students on so many levels. This post was about the challenges of bringing colleagues around to that perspective. My next post will cover more about the details of the class, conveying the student experience and discussing the unexpected thing that happened along the way.
Laura Kornish is a marketing professor at the University of Colorado Boulder. She is currently serving as the marketing chair. For more information, see http://leeds-faculty.colorado.edu/kornish/ and http://laurakornish.com/.