A while ago, I blogged an invitation to the 18th annual Legacy of R.L. Moore and Inquiry Based Learning conference, to be held in Austin, Texas on June 25–27. That’s only a month away! To lead in to this event, I’ve asked a few members of the organizing committee, who are leading practitioners in inquiry-based learning, to join us here for a 4+1 interview. The first of these is Victor Piercey. Victor is an associate professor in the Mathematics Department at Ferris State University in Big Rapids, Michigan (practically a neighbor to me, in other words). He took time out from the hectic pre-conference organizing schedule to answer a few questions about IBL and the R.L. Moore conference.
1. In your own words, how would you describe or define inquiry-based learning to someone who had never heard of this concept?
To me, inquiry based learning is all about engaging students in mathematical sense making activities.
Most of us (probably all of us) who teach college mathematics want our students to apply higher-order thinking skills to math. Students tend to memorize problem templates and then try to match each problem they face with the appropriate memorized template. Most of us who were successful in math learned that this is an ineffective approach to learning. For one thing, when you understand how multiple ideas are related and you understand where they come from, we can drastically reduce the amount of memory needed to learn math. Moreover, we truly retain what we learn in the long run, rather than delete the memorized templates after the course is over.
But there is much more at stake than how well we learned the mathematical content. While there certainly are some students who need to be able to use the content professionally, such as engineers, most of the students I teach in general education will never need the content I teach either in their lives nor professionally. How does solving an equation involving radicals alter your worldview or help you understand yourself?
In my estimation, the purpose of general education mathematics is to guide students through thinking clearly about problem solving in complicated scenarios. This goes beyond mathematical problems. In our lives we are often confronted by complicated, interconnected problems. For example, Michigan recently had a proposal on a ballot to increase the sales tax from 6% to 7%. On its face, deciding how you vote on such a measure is complicated. What does the state want the extra revenue for? Is this a good source of revenue or are their better avenues? What is the political likelihood of the state adopting other options? However, when one examines the purpose for this particular sales tax increase, the rationale involved funding for fixing the roads, funding for education, and the fuel tax (among other matters). At its heart, the entire measure was a “work-around” to fund roads and bridges. But it was very complicated!
This is why inquiry-based learning has such value to me. By engaging in sense-making activities, students develop healthy problem solving skills. They learn to listen to one another and look at the same question from multiple angles. A student who successfully gains all that IBL has to offer in a mathematics class can look at the Michigan sales tax proposal and put together a coherent picture of what the state is asking for, weigh the politics of the measure along with alternative possibilities for solving the road problem, and come to a conclusion based on sound reasoning.
2. In which of your own classes do you use IBL, and what does IBL look like when being used in one of your own classes?
I use inquiry-based learning in a course called “Quantitative Reasoning for Business.” This is a general education course that I invented. The idea behind the course is to prepare students for the kinds of mathematical thinking and reasoning necessary for success in business. The term “quantitative reasoning” strikes many mathematicians as a weak version of mathematics that avoids algebra. I do not take that perspective. We work with algebraic representations of functions in a very rigorous way, but based on the meaning behind the symbols. For some portions of the curriculum, I had trouble finding sufficiently challenging exercises across the entire Pearson library!
The course materials consist of activities I call “explorations.” These typically involve much more guidance than one sees in upper-division IBL courses. My course, being designed for freshman who are highly math anxious, provides a bit more hand-holding. Students work through a series of questions that leads them to mathematical conclusions. For example, near the end of the first semester, students are asked to figure out how much money they have to deposit in order to save up to a fixed target amount in a fixed amount of time (given a fixed interest rate). They know the “future value formula” and as a result are forced to solve an equation. Most students will substitute the numerical values and solve the resulting equation. The next question asks them to solve the same problem about 20 different times using data in a spreadsheet. This forces the students to manipulate the formula itself and then program that formula into Excel. This leads students to an understanding of the purpose behind formula manipulation. Later Explorations involve the use and meaning of inverse operations in the manipulation of symbolic formulas.
This is not to say that students are never asked to complete open-ended problems. However, I use these as learning experiences and introductions to topics. As an example, the very first exploration in a unit on linear systems and linear programs poses a complicated problem for students to play with. A restaurant that they manage is offering steak and a vegetable lasagna specials over the weekend. We have cost and price data along with budget and storage constraints. The students are asked to explore how many of each special they should be prepared to sell in order to maximize profits. While working on this Explorations, different groups of students discover different parts of a linear programming problem. Some students focus on the constraints and derive inequalities. Others focus on the profits. The following day, we hang linear programming “vocabulary” on their discoveries, and proceed to develop the necessary skills to solve the problem. Three weeks later, we return to this problem and the students solve it.
3. Do you have a favorite success story from one of your own experiences using IBL — or a failure story where you used IBL and it didn’t go well, but you learned from it?
I have several success stories, but one of my favorites involves a student who had a history of failing mathematics classes. This student took my course and ended up asking the most insightful questions I have ever heard. He was particularly fascinated by the idea of slope as a rate of change. During a task when students were to determine whether or not a table of function values could be modeled by a linear function, he asked how one would find the slope for a nonlinear function. Nonlinear functions must have rates of change too, he reasoned. He then offered a suggestion – he thought we should average the slopes around the point. One cannot help but be impressed with a student who could not pass a standard Intermediate Algebra course but could raise questions that open up to the very heart of differential calculus!
4. You’re involved with the Legacy of RL Moore conference. What can a person attending this conference expect in terms of professional development and the overall experience?
Somebody attending the Legacy of RL Moore conference can expect a great deal over a short period of time. First and foremost, one should expect great and engaging presentations! Somebody attending the conference without a background in inquiry-based learning should expect to see several examples of this pedagogy in action, and they should expect to leave with a great many questions. All of us should expect to question our own past and present teaching habits. For some, this can be a painful process, but the end result is joyous! Finally, somebody attending this conference will leave with many new friends who are eager to advise and help anybody grow both personally and professionally.
+1: What question should I have asked you?
A good question for me would have been “What have you learned through using IBL?” I have learned a lot about myself, about my students, and about mathematics. Given that I focus my attention on general education, the last may be surprising to some readers. After all, what can somebody with a Ph.D in math learn by teaching basic algebra? Quite a lot, actually!
My favorite recent mathematical insight has to do with inverses. I was trying to figure out how to help students understand why when multiplying/dividing both sides of an inequality by a negative number we have to switch the direction of the inequality, but not when we add or subtract. What I came upon was a geometric representation: Multiplying/dividing by -1 results in a reflection on the number line over the origin, which reverses the ordering. However, addition or subtraction results in a translation to the left or right, which preservers the order.
This can be taken further. A similar statement can be made about taking reciprocals, but this time we reflect over the number 1. I guess this is a reflection in the sense of reflecting over the unit circle in the complex plane. Then I saw something a little but deeper here. How do we even find additive or multiplicative inverses? For addition, we multiply by negative one. In other words, we apply the next operation down the line with -1 (thinking of multiplication as arising from repeated addition for whole numbers). The geometric result is to reflect over the additive identity. For multiplication, we raise the number to the power -1. In other words, we apply the next operation down the line with -1 (now we are thinking of exponents as arising from repeated multiplication for whole-number exponents). The result is to reflect over the multiplicative inverse. So the more general picture here has to do with operations forming a hierarchy based on repetition. We begin with addition, then multiplication is next and comes from repeated addition, and then exponents arising as repeated multiplication. One obtains the inverse element for an operation by applying the next operation using -1 and the result is a reflection on the number line over the identity under the original operation. This can even be applied to functions and composition.
I am sure I am not the only person who has figured this out. It may even be known by everybody else besides me! However, that does not diminish the joy of learning something new. This is something I would never have learned had I not used IBL.
One coda to this story is that I have a question that I have not yet figured out an answer to. If we try to think of exponentiation as a binary operation on an appropriate set of real numbers, we have to different inverse operations: roots and logs. I don’t have a precise answer to why this is the case, but I am sure it must come from the natural asymmetry in the operation. Not only does the operation fail to commute, but the two arguments in the operation come from different sets. The base comes from (0,infty) while the exponents come from the whole real line (I am including 1 as a possible base since we can define 1^x for any x, but not zero since 0^0 is indeterminate). Moreover, exponentiation isn’t even associative. Somehow I think all of this combines to tell a story regarding why we need two different inverses for exponentiation, and I think this story also has something to do with mathematical transcendentalism. I have not put all of this together yet, but I look forward to doing so at some point soon!