Flipping Math Is More Fun If You Just Do This

Lead Features February 20 / February 26, 2020

 – Peter Santoro –

The first thing most people think about when they hear Flipped Classroom/Flipped Learning is the video lessons. The most common misconception is that it’s all about the videos. They couldn’t be more wrong. The video lessons allow teachers to “create” more time with their students in class. I like to think of this additional time as a blank canvas that I can “paint” with active learning strategies and strategic activities that will build on the knowledge my students have accumulated from prior video lessons as well as prior knowledge. 

Getting creative

I have a colleague who is a special education teacher and co-teaches 10th-grade English. Her video lessons are geared toward special education students, but she makes them available to the general education students as well. In some of her video lessons, she reads short passages and instructs her students to read along with her. After she reads the short passage, she demonstrates how to find the main idea(s), and she helps them work on strategies to improve comprehension. This allows her to use the additional class time to provide one-on-one or small group instruction to struggling students with reading deficiencies. Also, her students always have the video lessons at their disposal should they need extra reinforcement of concepts when working on their own.

This same colleague also co-taught Earth Science five years ago. The majority of the video lessons were created for course content. Additionally, students received video lessons on lab procedures, which saved time when students were fulfilling the lab requirements for the course. This enabled the students to be able to do more hands-on lab work which reinforced, in a more tangible way, the course content. When they analyzed the state exam results for all students in the school, the students in these flipped Earth Science classes outperformed the students in the traditional classes.  

Mixing it up

In my math classes, I use the extra class time in several ways. I have always been a big advocate of using the “visual” to show the concepts more concretely. I use the free website www.desmos.com to help my students “see” equations and functions in a clear manner. I have Chromebooks in my class, and I have my students work in small groups. One math topic that is particularly suited to this app is the graphs of trigonometric equations. Before I introduce trig equations, I have my students go through a series of “self-discovery” exercises where they graph a variety of trig equations. They sketch the graphs and then answer questions about the behavior of the graphs when different parts of the equations change each time they graph a new equation. When my students connect the various parts of the equation with how the graph changes, it makes a once-daunting concept much easier for students to understand. I use this app when I have students explore quadratic equations, as well as equations of circles, ellipses and hyperbolas. Connecting the visual to the math is a game-changer for most students. Students can use this app at home as well, so being able also to use it when studying benefits my students in a big way.

Low tech, high creativity

Another activity I use in my math classes is something I call a “silent debate.” I use this when there is a multi-step process that my students need to master so that we can explore more complex concepts. The way this works is that I pair my students facing each other. It is called a silent debate because the students cannot speak…the entire process is without verbal communication. I give one student in each pair a problem. That student has to do the first step in the process, then they pass it to their partner who must do the second step, then passes it back. This continues until that pair finishes; then I give them another problem. If one person makes a mistake and their partner finds it, it must be corrected, then passed back to the have the other student do the next step. They cannot communicate what the mistake was; they must correct it, and then move on. This forces each student to be attentive to their partner’s work and know the process involved. I have found that many students do not do a very good job of paying attention to each other enough to be able to learn from each other substantially. This process teaches them how to be more engaged with each other.  

An app for that

Another app that could be useful for struggling students is www.deltamath.com This app gives struggling students problems to practice and gives them immediate feedback. They get similar problems in succession to reinforce the process for solving similar types of problems. This is similar to my “silent debate,” but students can use it on their own if they feel they need more practice. Teachers can assign Delta Math homework or use it in class. Since my classes are Flipped Mastery Classes, an app such as Delta Math can help students master individual skills and allow them to proceed to the next lesson. 


Another great activity that I have implemented with the extra class time that Flipped Learning allows is Project Based Learning. My colleague, Dan Jones, wrote a book on Project Based Learning in a Flipped Learning class, which is a great resource for anyone interested in PBL. The how-to that Dan provides took away any hesitation I had about implementing PBL in my classes. Many resources are available for project ideas across all disciplines, grade levels, and student abilities. One site I have used is the Buck Institute for Education/PBL Works site. It is free and searchable by grade, subject, etc. 

I have found that my creativity has grown in the years since I implemented Flipped Learning. My students have thrived and done well. I can cover more content in-depth as never before. I am happier as a teacher and my students are learning, applying their knowledge, and are more intellectually curious than my students were before I flipped my classes.

Do you see a typo or an error? Let us know.

Peter Santoro
Peter Santoro
I have been teaching High School Mathematics for 12 years. This is the fifth year I am “Flipping” and my third year with Flipped Mastery. In addition to two sections of Introductory Calculus, I also teach one section of Geometry and two sections of Mathematics Research Honors. In addition, I am the coach of the Garden City High School Math Team (Mathletes). I am a Founding Member of the FLGI International Faculty as well as an FLGI Master Teacher and a member of the FLGI Insanely Smart Panel on the innovative uses of class time.

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1 Comment

on February 28, 2020

Hi Peter
Thanks for sharing!

One additional technique that I learned from the Flipped Learning video courses is Peer Instruction. I start many of my math classes with a set of 2-4 problems that are challenging- it was recommended that 50% of the students would get them right. I put these on the projector. I have the students try to work alone, but eventually they will start talking in their groups. After I feel like they have talked about all of the problems, I have them get up and find someone else to explain their solutions to. The discussions at this time are terrific! If they didn’t know before, they will know now. If they picked it up from someone in the first go round, they now have to explain it. If they still don’t get it, I can work with them later.

Then I have a few students summarize to the whole group. It was exactly what I looking for to go deeper with topics and get more understanding when I started to flip many years ago.

The types of questions I ask:
1) Interesting problems that take them deeper or they discover on their own.
2) Problems that cover misconceptions.
3) Problems that may have historical importance.

One of my favorite examples is to have students discover Euler’s Theorem and Identity using sequences and series in Calculus. The oohs and aahs are fantastic!

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