I have also, over the past 8 or 9 years, been developing a different teaching approach using the “flipped classroom” model. Under this model, most of the lectures are outsourced to videos which students watch before they come to class, and much of class time is spent by the students working on problems in groups, and most of my time talking in front of the class is me directly responding to student questions by solving problems on the board. The idea behind the flipped classroom model is that class meeting time is the only time during the week when students are able to directly interact with their instructors, and that in-class time is better spent on activities in which student-teacher interaction is more effective, while lectures which require little interaction from the students are better completed outside of class.

In the Fall of 2022, I conducted a small classroom experiment. I had been assigned to teach two sections of Math 100, one after the other, and I decided it would be worthwhile to teach one section using each instructional model to see which one was more effective. In my pre-2020 teaching at Alexander College I almost exclusively taught using the lecture-based model, however, when lectures moved online in 2020 due to the pandemic, I adopted the flipped classroom model to better reach those students who had internet connectivity issues that prevented them from taking full advantage of synchronous online lectures. I felt that the transition back to campus was a good opportunity to compare these two models to see which one would be a more effective approach for my particular set of students.

Both sections of Math 100 in the Fall of 2022 were evaluated in the same way. Both had the same number of open-book quizzes, and the same number of midterm tests, and both wrote the exact same final exam. Both completed the same assignments, and both sections were given links to the same free online textbook. Both sections were given links to the same set of video lectures I had made, although only the flipped-class section was explicitly asked to watch the videos as homework. Both sections were graded on in-class participation in solving practice problems. However, the lecture-based section had less class time allotted to solving practice problems than the flipped-class section.

In the lecture-based sections, about 75% of class time was occupied by my lectures, with the other 25% devoted to students solving practice problems during class time. These practice problems were presented immediately after the material that they were related too, so students had an opportunity to see me solve an example immediately before they were asked to solve one themselves.

In the flipped-class sections, I reduced the time spent lecturing to about 30% of class time. These lectures were summaries of the video lectures, rather than full lectures in themselves. Another 40% of class time was devoted to students solving practice problems. About 30% of class time was students working on problems in groups, while 10% was devoted to games such as Kahoot. I presented these practice problems all at once rather than after each portion of the lecture, so that I could free myself up to circulate around the class and help students with those problems. The final 30% of class time was spent by me going over solutions to problems students had been asked to complete. I did this both for the practice problems I had asked students to complete in class and also for the problems from the weekly assignments.

One of the things I noticed through direct classroom observation was that the flipped-class students had much more difficulty solving the problems in class than the lecture-based students did. I believe one of the reasons for this was that the problems presented in the flipped-class section were genuinely harder problems than the practice problems given to the lecture-based section. The intention was to give students a familiarity with exam-difficulty questions to prepare them for the midterms and final exam. The hope was that the experience of solving harder problems in class would better prepare the flipped-class students for the final exam.

I also believe that another reason for the greater ease with which the lecture-based section completed their practice problems had to do with the way in which class time was organized. Since the practice problems came immediately after I had solved similar examples, a lot of students were able to solve the practice problems quickly and easily by simply copying the example I had just done. Students trying to succeed through ‘rote learning’ where they solve problems by copying an example is something I have observed as a common pitfall that students at Alexander College succumb to. Many students do not grasp the difference between true mathematical understanding, and the ability to get the right answer by copying a previous example. I hoped that, the flipped class students could learn better how to truly understand what the problems were asking for by having a separation between my examples and the practice problems.

I also noticed a difficulty staying on task in the flipped-class section. Many of the students would take a 10-15 minute break when given a 5 minute break, and when given time to work on problems in groups, would spend that time talking about non-mathematical topics. Many students would complete only 1 or 2 of the 5 problems they were assigned, and many would seem to be completely unfamiliar with the material when they came to class time, suggesting that they had not in fact watched the assigned videos.

I will also remark here that, in the flipped-class section, I did not receive as many request from students for help as I would have liked. I spent a great deal of the time I had allotted to helping students with their practice problems waiting for students to ask me for help. Students would ask their friends for help, but that help would usually simply amount to copying their friends’ answer rather than the friend actually explaining to them how to solve the problem. I believe that many of my students, perhaps due to the academic culture of their high school studies, or due to their difficulties with the English language, were not confident asking me for help when they needed it.

At the end of the semester, I tabulated the students’ grades from the two sections to get quantitative results I could use to compare the two approaches. I computed the students’ average grade (out of 100) for their homework, quizzes, and participation, and recorded their grades on each of the two midterms and the final exam. I then computed the mean, median, and standard deviation of each grade category for each section.

Grade Category | Lecture Mean | Lecture Median | Lecture St. Dev. | Flipped Mean | Flipped Median | Flipped St. Dev. |

Midterm 1 | 54% | 49% | 29% | 51% | 52% | 23% |

Midterm 2 | 46% | 41% | 33% | 40% | 37% | 29% |

Final Exam | 47% | 50% | 33% | 37% | 37% | 24% |

Homework | 77% | 83% | 28% | 83% | 92% | 28% |

Participation | 63% | 63% | 30% | 66% | 71% | 25% |

Quizzes | 58% | 63% | 33% | 56% | 61% | 26% |

Final Score | 53% | 54% | 31% | 49% | 50% | 24% |

One thing I will note at this point is that part of the reason for the higher standard deviation in the lecture-based section was due to the fact that the lecture-based section had 7 students who did not write the final exam versus 5 students in the flipped-class section. There are always a number of students who stop coming to class If we remove these students, the higher mean and median scores in the lecture-based section becomes even more apparent.

The flipped-class section had higher scores on homework and participation than the lecture-based section. Both of these are likely due to the greater amount of class time devoted to students solving problems on their own and in groups. The greater amount of time devoted to practice problems indicated to students that these problems were important, so more students attempted them. Also, students were instructed to start working on their homework assignment if they finished the practice problems early, which likely contributed to the higher homework grades.

At the same time, the scores on the quizzes and exams were notably higher in the lecture-based section than in the flipped-class section. It seems clear, that despite my intention to use in-class practice problems to prepare students for tests and exams, students in the flipped-class section were not as prepared for their tests and exams as students in the lecture-based section. There are a number of possible reasons for this which I will outline below.

One possible explanation is that the students in the lecture-based class simply took better notes than those in the flipped-class section, and because they had better notes, they had an easier time students for tests and exams. Most students are not used to taking notes from videos, while taking notes from an oral lecture is a skill most students are familiar with. Moreover, many students in the flipped-class section appeared not to be watching the videos I had assigned, and therefore were not getting the benefit of any lectures at all.

Another possible explanation is that the students in the lecture-based class took the class more seriously than those in the flipped-class section. Due to the amount of time spent by students working at their own pace on practice problems, and the fact that many students were able to start their homework assignments during class time, I believe that many students in the flipped-class section may have felt that they didn’t need to do much studying outside of class. Rather than spending time outside of class watching the video lectures I had assigned and then completing their homework assignments, I believe that many students skipped the videos and solved most of their assignments in class, leading them to spend less time on the course than other students. This meant that, in the end, they were simply less prepared for the tests and exams.

A third explanation I will propose is that the difference in test and exam scores between the two section is *not* in fact due to the different teaching model used, but is due to a pre-existing difference between the two sections. I have noticed that, when I teach two sections of the same course with one section in the morning and the other in the afternoon, the afternoon section tends to perform better in terms of grades. I believe this is due to the priority registration system used at Alexander College where stronger students get first pick of which section to register in, and that afternoon sections are generally more popular. The flipped-class section was a morning section while the lecture-based section was an afternoon section. I would need to compare two morning sections or two afternoon sections to eliminate this explanation.

While it is possible that the difference in grades between my two sections was due to a pre-existing difference between the two sets of students, evidence still suggests that my lecture-based approach is the more effective one under the current circumstances. This is not to say that I will completely abandon the flipped-class approach, but that I will need to rethink the approach to ensure that students take my flipped-class sections as seriously as they do my lecture-based sections. This may mean requiring students to write online quizzes to prove that they watched the videos, or creating a more structured format for in-class activities. While I hope to test out a more effective flipped-class format in the future, for now I am used the lecture-based format for my in-person classes.

]]>In Fall 2022, I had the lowest ever attendance rate I have had in my time teaching Math 099. The last few weeks of class, I had an average of only 4 students attending each class out of a total of 30 students registered. While the number of students who wrote the final exam was significantly higher than that, the passing rate was dismal. Clearly, what I was doing – which was generally the same as what I had been doing for the past 5 years – was not working. It was obvious that I needed to do something differently.

The inspiration for the specific changes I made to my Math 099 course came from the Fall 2022 Professional Development Day at Alexander College. During that PD Day, the keynote speaker was “Anything but Soft: the Value of Transferable Skills in Acion” by Candy Ho. The objective of the presentation was to help faculty at Alexander College better quantify the transferable skills (meaning non-subject-specific skills such as time management, presentation skills, writing skills, etc.) that students are learning in their courses. However, the thing that struck me in the presentation was mainly that my Math 099 students were missing most of these skills. They had a hard time coming to class, had a hard time staying focused during class, seemed to never study outside of class time, and when they did study they seemed to only do it in ineffective ways.

This led me to my hypothesis: that the reason my Math 099 students were performing poorly – and the reason they had always performed poorly – was that they lacked a number of these transferable skills. Since I wasn’t teaching them these transferable skills, they were never able to succeed in my course, even if they took the same course over and over again. In other words, the problem I had in Math 099 was that I was dealing with students who had never properly learned how to learn.

Thus, this semester in Math 099, I dedicated the whole first week of class to trying to teach my students how to learn math. While there was a part of me which resisted giving up time that I could be using to get ahead on the course syllabus, I knew that spending more time covering the material was not going to help my students. I needed to teach my students how to get more out of the class time they had, rather than giving them more of the same.

Before this winter, the first class of Math 099 was spent reviewing the contents of the course syllabus: going over where to get the textbook, assignment instructions, where to find things on the course Canvas page, attendance policies, etc., and then during that first class, students would play a Kahoot! quiz game to test their knowledge of the prerequisite material. Then, in the second class, we would jump right in to reviewing that prerequisite material (in the course syllabus I call this Unit 1: Arithmetic Review).

This winter, I spent the whole first class doing a rather involved interactive Mentimeter presentation. This presentation included most of the topics that I previously would have covered in the first class, but also included a number of additional topics. These additional topics included:

- The percentage of students who failed the course in the previous semester
- Various reason
*why*students had failed in the past (low attendance was at the top of the list) - How many hours per week students were expected to spend studying
- The way in which mathematical knowledge builds on previous knowledge and its implications (if you fall behind it is really hard to catch up)
- Mathematics as the study of patterns
- Why practice is important when learning mathematics
- The ways in which all of the assigned in-class activities and homework were designed to help students practice

I felt that students were much more engaged in my first class presentation this semester than they had been in previous semesters. I know part of it was because of the interactive elements of the Mentimeter presentation (normally I present by writing in a OneNote notebook on my tablet PC that is projected onto the whiteboard – this is very versatile, but has few ‘gimmicks’). Part of it, though, was also because, when I was explaining how the course was going to work, I was also explaining *why* I had set up the course in the way I had. I tied every element of the course to the ways in which it would help students study and learn.

In my second class, I re-used the same quiz game on prerequisite material that I has used to give students in their first class. I wanted students – especially those students who hadn’t studied math in years – to have a good idea of where they were starting from in their learning journey. However, rather than moving straight from there into reviewing the prerequisite material, I next moved into a short Mentimeter presentation. While the first class presentation was focused on *why* studying every week was important, this next presentation was focused more on the difference between ineffective and effective studying. I tried to convey the idea that memorization is a very ineffective way to study math, and that the purpose of repetitive examples is not to memorize the steps used in those examples, but to learn a general rule that can be applied in novel examples.

The final activity in my students’ second class was the creation of a study plan. I asked each student to make a plan for their weekly studying for the semester, and either email it to me or post it in an online discussion forum. On this study plan, students how to indicate:

- How much time they planned to spend studying each week
- What their most important at least important activities were for out-of-class studying
- How much time they would spend on each individual activity
- When during the week they planned to work on each individual activity

It was then only in the students’ third class, at the beginning of week 2 of their course that we actually began working on the material from Unit 1. Even by week 2, I was already seeing a marked difference in attendance and engagement from last semester to this semester. While there could be other explanations for this difference, I hope that the time I spent on helping my students learn how to learn will help them throughout the course and beyond. I hope to make another blog post towards the end of the semester keeping readers informed as to what effects I saw from these changes to Math 099.

]]>“Why now?” you might ask. “Why is Telyn choosing this moment to relaunch eir blog?” Well, I have made a decision to go back to school for my Ph.D. in Math Education. I made the decision that I wanted to *eventually* do this over 5 years ago, but the time was not right until now.

As I have been reading up on Ph.D. programs and the application requirements, one of the things that has been standing out to me has been the requirement to turn in an academic writing sample. The truth is that it has been many many years since I have done any academic writing. I have not taken a course or written a paper in over five years, and I simply do not have anything to include as part of my Ph.D. application. It also seems a shame to me to start writing again, but to know that my writing will not be read by anyone other than university admissions officers.

This means that the primary purpose behind me relaunching my blog is to get practice writing again. I wish to exercise my writing muscles through this blog so that I can practice for my future studies. However, this blog is going to be more than that. Through researching various Ph.D. programs, I have realized that I am not certain where my current research interests lie. I know certain areas of education research that interested me years ago when I was taking education courses. It has been long enough since then that some of those interests have waned. Moreover, I simply am not up-to-date as to current developments in the scholarship of Math Education.

Therefore, part of this blog is going to be me commenting on reading I am doing into math education. I will use this as a way to challenge myself to perform independent study into some of the areas in which I am interested. However, I will also be using this blog as a way to talk about what is going on in my own classroom. Us Post-Secondary educators rarely get the opportunity to talk to anyone beyond our own departments about what we are doing in the classroom. I hope that, through this blog, I can help others learn from my teaching experiences just as I will be learning from the research of others.

I look forward to writing this blog, and I hope that others can get as much from reading it as I get from writing it.

Till next time,

Telyn

]]>5.1.1 Inverses (7.5 min)

5.1.2 Inverses and One-to-One Functions (10 min)

5.1.3 Finding Formulas for Inverses (18.5 min)

]]>I was inspired to write this post as I was riding the bus back from the CanFlip14 Conference in Kelowna. I came to the conference to learn more about the “flipped classroom”, an innovative model of instruction in which lectures are replaced with videos that the students watch at home and homework is replaced with interactive activities that happen during class time. While I did learn a fair bit about the “flipped classroom” and other innovative approaches to instruction, the bigger thing that I’m taking home from this conference is the importance of having a PLN (for those who, like me when I arrived at the conference, have no idea what a PLN is – it is an acronym for “Personal Learning Network” – a group of people who you share ideas with in order to help you learn more about how to improve your own teaching). I have learned that if I am to work on improving my teaching and introducing new ideas into my androgogy (I choose not to use the term pedagogy as my students are NOT children), it is much easier to do it in a community than in isolation.

Working in isolation has been how I have spent much of my teaching career. Some of this is due to the academic cultures in which I have worked which have usually valued autonomy over community. Some of this is due to my own experience studying in research-intensive universities in which faculty didn’t really talk to each other about teaching, because they had used up all their social energy talking to each other about their research. And some of it is due to my own fears that, as someone who advocates a very non-traditional approach to university-level education, my teaching innovations wouldn’t be supported by my colleagues. I’m not sure if this is a “grass is always greener” situation, but I sometimes feel that post-secondary math departments tend to take a more traditional approach to curriculum and teaching methods than other departments do.

But, having met other folks at the conference who are working through some of the same things that I am, I am feeling more and more that there are people out there who I can reach out to and who I can learn with. They may not work in the same institution as me, or be teaching the same level of material as me, or be teaching in the same discipline as me, but they have good ideas that I can learn from and I have good ideas that they can learn from. The internet can be a tool not only for building social connections, but also for building professional connections which can give me access to a community so I do not have to break my own trail all the time.

When I think back, it was years ago that I was told that I should create a teaching blog. Back then it was in the context of looking for teaching work and the idea that by sharing my ideas in a blog I could help show prospective employers what I stand for and what I work towards in my classrooms. But, even since I have been regularly employed I have had the same suggestion made by colleagues who felt that I had good ideas to share and believed that having a blog could be a great way to share these ideas with the world.

So, now I have resolved to repurpose this web site to serve as a home for my teaching blog. No longer will the “posts” section of my webpage be solely limited to me making announcements about my tutoring business. I have resolved to make regular posts sharing my ideas an experiences about teaching. I will share my sucesses and failures, my dreams and fears, and hopefully find that there are other people out there who can be inspired by my ideas and others who have suggestions to help me move forward. But, if I don’t put any of this stuff out there, then my ideas don’t have the potential to meet other people’s ideas, to mate and to give birth to litters and litters of wonderful new idea-lings!

So, to make a long story short, welcome to my new blog!

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Also, happy international women’s day to all you women and girls out there!

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