To show calculus students why limits are important, Ondrej Zjevik, MS, presents them with a problem that they cannot solve with the math they already know.
Instructor of Mathematics, Florida International University in Miami
MS in Mathematics, MS in Applied Mathematics with a minor in Computer Science, BS in Computer Science – Computational Mathematics
“Give me a problem, and I want to solve it,” says Ondrej Zjevik, MS. That is a good thing because, as a math instructor, he faces problems—quite literally—every day.
That desire to solve problems has taken Zjevik far. In 2012, he left his home in the Czech Republic to study at the University of Minnesota, Duluth, where he earned his master’s degree in applied mathematics—or, rather, his first master’s degree. His second MS in math came from the University of Miami. He now teaches a variety of courses at Florida International University (also in Miami), in the Department of Mathematics and Statistical Sciences.
Of course, not all students in Zjevik’s courses are as enamored with math as he is. Many in his Calculus for Business course, for instance, are business majors who have not retained their algebra skills. “I have to refresh essentially everything that is covered in Algebra II,” Zjevik says. “And I have to refresh it every other class.” In fact, students’ ease with math runs across all levels: About one-fifth of the students need help with things as basic as simplification of fractions, he says, while others say, “Wow! This is really easy!”
One area in particular in which students struggle is limits. When certain types of problems cannot be worked out directly, mathematicians use limits to plug in numbers that get them closer and closer to the answer. The limit of a function describes how that function behaves near a point, rather than at the point. It is a challenging concept but one that students need to understand before they delve deeper into Calculus for Business.
All of this presents Zjevik with a problem that is challenging to solve. Luckily, that is just the way he likes it. And, not surprisingly, he has arrived at an effective solution—one that begins with flipping the usual approach to limits upside-down.
Students have zero experience with limits
Zjevik has found that teaching limits without saying why the tool exists makes it difficult for students to understand why it is used. “My opinion is that teaching ‘how’ without ‘why’ is not effective,” he says. “The students will forget most of the methods and procedures we teach them in math, but if they know the reason why is something done a certain way, they might recall the methods faster.” He wanted to find a way to get students to understand the “why” behind this type of equation and, at the same time, become excited about solving problems.
Open students’ minds with an “unsolvable” problem
After reviewing algebra, the first things covered in Calculus for Business are differentiation and limits. Zjevik says that the way limits are usually taught is by “starting with the boring math first, with the math language.” But he has found it works better to have students dive in with a strong example—something that shows them that none of the techniques they have learned will work. This provides the epiphany that gets students’ brains working, so they are actually excited to learn the tools they need.
“If you understand it, math is so beautiful—everything is connected with everything else. When you see the connections, that’s the aha! moment. [Students] realize that is why we have the rules. Everything fits together like puzzle pieces.”— Ondrej Zjevik, MS
Course: MAC 2233 Calculus for Business
Course description: Basic notions of differential and integral calculus using business applications and models including: differential and integral calculus using polynomials, exponential and logarithmic functions.
See resources shared by Ondrej Zjevik, MSSee materials
Lesson: You cannot divide by zero
As mentioned, Zjevik begins the lesson on limits and differentiation not with a lecture or instructions, but by giving students a problem and asking them to work together in small groups to solve it. The problem Zjevik gives them is, he says, “really, really hard or impossible to do with the rules they know.” (More on that, below.)
Zjevik lets the students struggle for a while to see what they come up with. The students do not realize this is something new, so they attempt to do it the “old” way, using what they have been taught in math classes up to this point. They try algebra, but that does not work. They soon come to realize that none of the rules they have already learned will solve the problem.
After students hit this roadblock, Zjevik teaches them the new rules they can apply to problems like this one. Therefore, the activity serves both to reinforce the old rules (by testing them out on a new type of problem) and to show where the old rules cannot be used. It also gets the students thinking critically and collaboratively and, in many cases, motivates them to pay attention to what comes next.
Here is how he presents the challenge:
Pose the problem
Zjevik provides students with these simple directions:
Think about stopping your car at your house. You get out and walk from the car to your front door. Measure the distance at both 2 and 5 seconds. What is the average speed you are walking?
Let us say the distances were 4 feet and 7.5 feet, respectively. To find the average speed, students have to find the difference of the distances and divide by the time walked. This is done with a fraction—the denominator is the time, and the numerator is the distance. So, the first walk would be 4/2 and the second would be 7.5/3, or 2 feet per second and 2.5 feet per second, respectively.)
Students should make it this far in the calculations with little problem.
Favorite Math Materials
Zjevik is always looking for ways to enhance student learning. He lets his students use Wolfram Problem Generator so that they can practice on a platform other than their homework. He also created a Chrome extension that tracks their progress so that he can give them credit for correctly answered problems.
Further, Zjevik initiated the use of a free homework system called IMathAS at Florida International University. “We are still in the early stages,” he says, “but this system is free to use for faculty and students at FIU and is used by 1,000-plus math students per semester. We plan to expand the system for the whole university in the future.”
Introduce the idea of limits
Next, Zjevik ask students a slightly different question: What if you wanted to know what your exact speed was at, say, 3 seconds or 4 seconds?
Students can tell from the initial figures that their speed picked up as they walked, so the average speed at 3 or 4 seconds will not be 2.0 or 2.5 feet per second. To get closer to the correct figure, the students will need to make some new calculations.
Zjevik then narrows it further: What would be the average speed between 2 seconds and 2.0001 seconds? As students consider the math here, the denominator is going to be really small and the numerator will be almost nothing, because they will have traveled very little distance in very little time.
Zjevik then explains to students that “limits” is a mathematical tool that helps make sense of this type of fraction.
Introduce the idea of instantaneous speed
Zjevik then asks the students how they can look at what is going to happen to the average speed when the time goes to zero. They realize that putting zero in the denominator does not work, because that is not a viable answer (0/0 is undefined).
After the students struggle with the problem for several minutes, they nearly all grow frustrated. Zjevik asks them what they are thinking. They tell him that it looks like the rules they have learned will not work in this situation. This is their moment of realization: They have not yet learned to solve the problem, but they now understand why they need a new tool.
Walk them through the solution
After the students realize that they do not have the tools necessary to do this kind of math, Zjevik explains how to use limits to make sense of the problem.
Here it is in practice:
Take them to infinity and beyond
After considering what happens when the time goes to zero, Zjevik has students look at what would happen if the time went toward infinity. He tells them that this type of problem also can be done because of limits.
Now (finally) put it into math language
Once students have tried the zero and infinity problems, Zjevik explains, “This is what limits do. Let’s see how we can put it in math language, and you have a whole new way of approaching problems.”
He tells them that there is notation for this work, and they have to be careful to choose when they can and cannot do the simplification. Limits say that, if you can simplify the fraction and then you can replace h with zero, the two fractions are, in some sense, equivalent.
Zjevik has found that students who are allowed to work first without direction explore and have more fun with the material than those who are told how to complete the assignment before they begin. He has seen increased engagement when using this technique, which increases motivation and understanding.
Students have come back to Zjevik after taking his course to say they are thinking about changing majors because “math is cool.” He agrees. “If you understand it, math is so beautiful—everything is connected with everything else,” he says. “When you see the connections, that’s the aha! moment. [Students] realize that is why we have the rules. Everything fits together like puzzle pieces.”