When Kristen Jaskie, MS, saw students struggling with critical thinking in her discrete mathematics class, she solved the problem—with puzzles.
Professor, Glendale Community College, Glendale, AZ
MS in Computer Science/Machine Learning, BS in Computer Science
Kristen Jaskie did not discover her passion for teaching in the halls of a high school or on a university campus. It happened in the mountains of Arizona—or rather because they were threatened when she was 13 years old.
That was when Jaskie’s mother read her an article about the McDowell Mountains, located north of Scottsdale, not far from their home. The story told of a group that had formed in 1990 to try to save the mountain from a land developer. Jaskie, who loved this natural monument’s light-and-shadow face and its winding trails, decided to help.
“I grew up playing in the desert and going on hikes with my dog in the desert around my house, the way some kids go to playgrounds. I knew the desert—both the beautiful parts and the dangerous parts—and I was horrified by the thought of them being devoured by fancy houses.”
She set up a doughnut sale at her school and raised $30. When she presented the money to the land trust committee, the directors were so impressed that they asked her to join the Board of Directors. As she worked with the trust over the next several years, she discovered that she loved public speaking. In particular, she relished returning to her former middle school to teach students about the importance of conservation and the biology and history of the Sonoran Desert
Moving from mountains to mathematics
Today, Jaskie has a bachelor’s and master’s degree in computer science and is working on her PhD in electrical engineering, with a focus on using artificial intelligence and machine learning in an electrical engineering setting. But she still indulges her passion for teaching, working as an instructor at Glendale Community College in Arizona. In that role, she frequently finds herself introducing college students to a topic as foreign as conservation was to those middle schoolers: discrete mathematical structures.
“Instead of continuous math, like algebra, trigonometry, and calculus, in which real numbers and decimals are used, discrete math deals with integers [whole numbers],” she explains. Discrete mathematics, she adds, is the foundation for understanding computation. “For example, the European space launch Ariane 5 veered off its path and exploded due to a conceptual computation error involving floating point values. The launcher had cost $7 billion to develop and was carrying $500 million worth of cargo. That error could have been prevented by a better understanding of computation. If you think about all the electronics in our lives today, these are all founded on these concepts.”
Jaskie says the subject covers sequences, functions, set theory, probability theory, and graph theory. These theories typically are not covered extensively in K–12 education, she adds, so students often struggle with them.
One particularly problematic concept: proofs. Proofs are important for students pursuing computer science degrees because, later in their coursework, they will need to use proofs to prove that their algorithms are correct, and they will have to lay out their arguments with no possible loopholes. “It forces them to think more rigorously and logically than in general math,” she adds.
Working on proofs requires a level of critical thinking that students have often not experienced before. While puzzling over how to help them succeed, she realized that the answer might be puzzles themselves.
“Discrete mathematics is the foundation behind computation. To understand how computers really work, computer scientists have to understand the mathematics and the logic behind them, and the algorithms and programs that run on them. As part of a computer science degree, students will have to come up with unique solutions (algorithms) to difficult problems. They will then have to formally prove that their solution will always work and why. Discrete mathematics is the first step in doing this.”— Kristen Jaskie, MS
Description: Introduction to lattices, graphs, Boolean algebras, and groups. Emphasis on topics relevant to computer science.
See resources shared by Kristen Jaskie, MSSee materials
Solving students’ math “problems” with puzzles
To get students’ minds ready for critical thinking, Jaskie begins every class with a puzzle or logic question. The puzzles “wake students up,” she says, and they allow her to introduce new concepts in a less intimidating—and, frankly, more enjoyable—way. Here is one example:
“This is the first thing they see when they walk into class on the first day,” says Jaskie. “Initially, many of them are confused. They usually don’t think there’s enough information to solve the problem and think I’m just trying to confuse them. Students then start guessing. Usually, one or two students will figure it out. When we go over the logic behind the answer, the students are surprised and generally quite interested.”
Introducing the “why” behind the puzzles
Jaskie introduces a logic puzzle in the very first class of the semester just to get students used to the idea. But after they attempt to solve it, she explains to them why she thinks puzzles can be helpful to them by showing the impact puzzle solving can have in the human brain.
For this, she uses slides that illustrate brain activity during various mental tasks. During a typical class lecture, for example, research into electrodermal activity by Poh, Swenson, and Picard has shown that there is very little brain activity—in fact, it is about as flat a line as when watching TV. This is in contrast to the active brain function that takes place when students are doing homework or lab work.
Drowning in a sea of practice problems?
Jaskie was overwhelmed—and her students frustrated—until she implemented the Five Tries approach.
To address paperwork overload from grading quizzes and assignments, Jaskie has her students work on Canvas, an online learning platform. She provides homework on paper, and students submit answers into the system, which tells them the problems they got wrong—but not which parts.
Their job then is to identify which problems they are confident they have correct and which they are not sure about. They then rework the ones they think they have gotten wrong. At any point in this process, they can use their textbook, work with other students, or ask for help from Jaskie during office hours.
Students are given five chances to get the right answer. By working through those five tries, says Jaskie, they become adept at solving the problems. Since implementing Five Tries, student grades have gone up, Jaskie adds. In the past, most of the students ended the semester with Bs or Cs. Now most earn As and Bs.
Jaskie then explains that active thinking is required in a difficult subject like discrete math—and that she uses logic puzzles to fire up students’ brains so that they are ready to learn. “People learn better if they solve problems in class rather than just in the homework,” she asserts.
By the end of the semester, students often say that the puzzles are their favorite part of the class. Jaskie finds this remarkable, because college students usually hate being told they have to work with their classmates. On the first day of class, for example, when she tells them to work with their neighbors, she often hears, “This is so stupid.” After doing a couple of the puzzles, however, their attitude changes.
“They start talking and joking around,” Jaskie says. “They get involved in the problems … and debate amongst themselves. It’s really fun to walk around the classroom while they are working on a problem. The vast, vast majority of the students are extremely engaged in the material.”
Jaskie’s steps to puzzle presentation
While Jaskie starts each class with a puzzle or logic question, she also uses a specific progression to help students solve the problem: The students work to solve them on their own, and then with a partner, and then as a class. (She says this technique is called peer instruction and was developed by Dr. Eric Mazur, a physics professor at Harvard.) Along with waking students up, this method helps Jaskie encourage collaboration—all while easing students closer to the math concept of the day.
Here is the sequence Jaskie uses for in-class puzzle solving:
1. Have them try it on their own.
When they think they have the answer, they enter that answer into an online program, called Socrative, that is like a more traditional “clicker” program.
2. Have them try it with a partner.
Jaskie says this “peer instruction” is a form of active learning. Students get together with their neighbors and discuss the solution and how they arrived at it. In this activity, each student tries to convince the other that he or she is right.
“Studies have shown that, by committing to their own answer, students are forced to come up with an answer, even if they just guess,” Jaskie says. “This primes the brain to be more open to the topic.” The students often engage in energetic discussions over how to solve the puzzle, she notes.
3. Give them a chance to change their answer.
After conferring with classmates, students reenter their (sometimes different) answers into Socrative. Three quarters of the students or more usually have the right answer at this point.
4. Work through it as a class.
Finally, Jaskie explains the solution and walks the class through how to find it.
It is not just Jaskie’s students who enjoy the puzzles, she adds. Her slide presentations (which include the puzzles) have become very popular in other courses at Glendale—and beyond. In fact, her husband once took a course at Scottsdale Community College with a teacher who was using Jaskie’s slides. That teacher, her husband found out, had gotten them from a professor at Arizona State University!
“It’s very flattering to know that so many other teachers in the area have a high opinion of my slides,” Jaskie notes. “I’ve certainly put a lot of work into them! A lot of the puzzles I use come from a book called The Lady or the Tiger? and Other Logic Puzzles by Raymond M. Smullyan. I also use puzzles from his other books, and books by my favorite mathematical writer Ian Stewart. Others I find online…. Some of my favorites include the Knights and Knaves puzzles from Smullyan’s books. They are very useful to teach logical thinking. Probability puzzles are probably my other favorite. I love working through … fun puzzles from the Internet, like this one:”
“We discuss it and work through it,” Jaskie says …
“ … before getting to the final answer:”