August 6, 2018
Student-Centered Mathematics: Structure of the Class and Tracking Student Progress
August 6, 2018
Student-Centered Mathematics: Structure of the Class and Tracking Student Progress
This post is the third in a series about my university’s novel required math class for business undergraduates. Our course is called MATH 1112: Mathematical Analysis in Business. The class is organized around applications of math in business settings, and students learn to work on quantitative problems using spreadsheets. The first post explains the goals of the course, and the second post describes the process of putting it in place. This post offers more detail about the course, focusing on the structure of the class, the activities that comprise the class, and some serendipitous benefits of the implementation choices we made.
Our old math requirements were stereotypical math classes. They used textbooks and followed a set of mathematics topics such as functions, logarithms and exponents, probability, limits, and differentiation. The new class is not typical. It is organized around business settings that are naturally infused with data.
The first module presents several scenarios that involve currency exchange. The data are exchange rates, including snapshots and trends; and cash flows, including expenses and revenues. What are the relevant mathematics in this context? Unit conversion is a central mathematical concept, percentage change is important, and solution concepts (what would the rate have to be…?) come into play.
Students vary in their fluency with these foundational skills. But we aren’t going to drill them on percentage change or unit conversion. That would be insulting to our top students, as they have been converting feet to inches, dollars to dimes, and gallons to quarts since primary school. It would be boring for everyone, even to the students who need it most. So instead of reviewing methods, we just drop them into the deep end of the scenarios and encourage them to swim to shore, knowing that they have all taken swimming lessons.
Your company incurs expenses in yuan and realizes revenues in dollars. Here’s the exchange rate between those currencies. Here are expenses and revenues. What’s the profit margin? What if the dollar gets 10% stronger against the yuan? What happens to the margin?
How can they tell if they are swimming towards shore? We built a technology-enhanced version of “check your answers at the back of the book” to help them gauge their own progress. Because all of the modules in the course are set in data-rich contexts, it is straightforward* to build question banks that let students practice.
*Aside: Building question banks is straightforward but a huge amount of work! We did most of it with code, but code needs to be checked, and when we update the data sets because, of course, exchange rates change, we check and check again. This amount of work only makes sense at scale. If only 50 students took this course, we wouldn’t do it this way. With 1500+ annually, the return is there. Part of my motivation for writing this blog series for Complete College America is to find other people to join this mission. The bigger the scale, the more these investments of time make sense, and the better the materials are for everyone involved.
Regarding the check-your-work and move-at-your-own pace vibe of the course: this is not new. (I have very fond memories of SRA cards in second grade. And the teacher not knowing what to do with me when the box lasted me a week instead of a semester.) Here’s how we implemented it in MATH 1112.
The course has three types of regular activities: in-class participation quizzes, homework, and module quizzes. There is also a final exam. All of these activities are fueled by a massive set of questions we wrote. All told, there are about 500 questions, each with 10-100 versions, across the ten modules in the course. This is a lot, but in iterating and improving the course, it’s still where we focus effort. Great questions are the lifeblood of the learning in the course.
The questions are like the ones given above for currency exchange, for example, what’s the profit margin [given exchange rate, expenses, and revenues]? They are not open ended; they have right answers. Some are multiple choice and test concepts more abstractly, but most have numerical answers to enter. Many of the specific choices we made about format were based on convenient technology tools, most notably, our school’s learning management system. (It was D2L; we are in the process of moving to Canvas.) In the 10-100 versions of each question, we vary inputs in the question. For example, one version has revenues of $150,000 and another $200,000. We try to keep the versions of a single question at the same level of difficulty.
That’s what we mean by a question and by versions. But what’s a great question? A great question tests something important and is not obvious to someone who is just guessing. A great question is also diagnostic. A great question distinguishes whether the student is making progress on the course learning objectives. For example, we can ask a sequence of questions that help us diagnose whether a student partially or (nearly) fully understands the principles of unit conversion, and where the understanding falters.
We use the bank of questions for the three regular activities (in-class participation quizzes, homework, and module quizzes). All of these activities are fully integrated with our learning management system (LMS), which means they are all computer-based and automated. I don’t think of that LMS integration as part of the “essence” of the course, but rather a practical reality that makes the course highly scalable. As with all things technology, there are pros (automatic grading) and cons (“I don’t have my charging cable!”). Each activity has a distinct role in the course.
• In-Class Participation Quizzes: Students answer one question. They get most of the points for simply answering and the remainder for a correct answer. Participation quizzes assess diligence and attendance and give students low-stakes feedback.
• Homework: Homework is due about four times per week, with an 11:00 p.m. deadline. Each homework has 3-5 questions. A student sees a random draw from the versions of each question. Students can attempt the homework as many times as they want until the deadline. Their score is the highest score of all their attempts. The homework schedule encourages students to keep up and to repeatedly practice to self-diagnose what kinds of problems are sticking points. Students might not be able to label the math skill they are stuck on (Order of operations? Unit conversion? Geometric mean?), but they might be able to see patterns, described in their own words, in what is challenging them. Homework assesses both mastery and good habits like persistence and gives them self-serve feedback.
• Module Quizzes: The course has 10 modules over a 15 week semester, so they have an in-class quiz every 1-2 weeks. Module quizzes assess mastery.
At the beginning of this post, I promised to tell you about the structure of the class, the activities that comprise the class, and some serendipitous benefits of the implementation choices we made. So now for the good part, unexpected benefits!
The unexpected benefit is that this course produces useful data for student success. We see a lot about what our first semester students are doing academically through the digital footprints they leave in the learning management system. We can tell if they are attending class (or at least attending with a functional computer) via the participation quizzes. We can tell if they have good study habits by whether they persist for perfect scores on the homework. And we can tell if they are achieving mastery by module quiz performance.
All of this data, though, isn’t helpful unless we have a way of coaching the students. Here’s where we depart from technology. Our first semester students are enrolled in a freshman seminar. These seminars meet once a week and are led by our academic advisors and other members of the student services team. Every other week, I pull reports on the three activities for those leaders about the students in their charge, and give the leaders specific instructions about who needs a conversation and what to say. For low participation scores, speak to the student about impediments to attending class or about computer issues. For high homework scores but low module quiz scores, explore what kind of help (or “help”) the student is getting on homework. For overall low scores, steer students to the resources for help in the course such as open office hours. (All students can attend office hours of all course instructors.)
Most of the seminar leaders love being able to give such personalized and specific coaching to their students. It’s one thing to provide blanket nagging to a group of students about keeping up with their studies. (Cue deathly stares of indifference.) It’s another thing entirely to be armed with information: “Marcel, I see you have 80% on your math homework. Did you know you can retake until you get 100%?”
• Oh, you didn’t know that?
• Oh, you leave it to the last minute? Maybe a bad idea?
• Oh, you have been sick with mono for two weeks? Let me help you reach out to all of your professors to make a plan to get caught up.
I am very much aware that this level of personal attention for our first-year students is an absolute luxury. Our undergraduate associate dean (Al Smith) has been the champion for these seminars and another one of my partners in maximizing the influence of this project.
On our broader campus, there is talk of “early alert” systems and discussion about what data could feed them. This type of course naturally produces useful data for those systems.
This math class was designed to serve the needs of our students as future business professionals, but in the journey of developing the class, I’ve noticed auxiliary benefits, the ways in which this class happens to serve other strategic goals of our university around student success. As I mentioned in my initial post, this has been a labor of love. What’s not to love?
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/.