Welcome to the Introduction of Mathematics Proof!
We want your experience this semester to be successful and rewarding. Math 102 is a challenging course that demands consistent hard work throughout the semester. Here are some tips for you to succeed in this course and some common mistakes you want to avoid.
• Expecting to be graded in the same way as high school
In a university level math course, your grade is based primarily on tests. You cannot pass this course without achieving passing grades on tests. The only way to do this is to master the skills and concepts through careful completion of the homework exercises, review of the textbook and class notes, and extra practice whenever needed. If you get a low grade on any quiz or test, you are in danger of not passing. See your instructor immediately for tips on improving.
• Mistaking recognition for mastery
Students think that because they’ve seen the material before, they “know it”. This can lead to laziness at the beginning of the semester. Many students wait until they get a poor grade on a quiz or test before they get serious about the course. By then, it may be too late. Work hard from the first day to avoid this. Remember, you only “know it” if you can do it. This means you must be able to write out correct solutions for every homework exercise without referring to your textbook or notes.
• Believing that with mathematics, you either “get it” or you don’t
This is a myth. Every student can have success in mathematics with enough hard work. How much depends on the individual’s background and experience. However, it is important to realize that you can earn the grade you want with sufficient hard work.
• Not setting aside enough time for homework
Many students are over-committed with work, school, and family responsibilities. Without time to devote to homework and studying you cannot learn mathematics. You must adjust your schedule to allow sufficient time for your math class. While there are some classes where you might be able to take shortcuts, mathematics is not one of them. If you don’t have a minimum of 15 hours per week to study outside of class, you are setting yourself up for failure.
• Misunderstanding how mathematics is learned
Learning algebra involves skill acquisition. It is analogous to the physical training involved in music and sports. You would never expect to learn to play piano by going to a concert two or three times a week. Likewise, you should not think you have learned some mathematics just because you went to class and understood your instructor. Your real learning begins when you try to do the homework exercises on your own. You have “learned” a section of material only when you can write out the solutions to all the homework exercises without aid from your textbook or notes.
• Not addressing lack of preparation
College Algebra is Pre-Calculus (without trigonometry). It is expected that you have a working knowledge of Algebra 2 from high school, or Intermediate Algebra from a community college. If you don’t, you must get to work immediately to fill in the gaps. There are many resources at your disposal to help you review. Use them! Your instructor will describe all of the available options.
First I have to thank one of my favorite podcasts: Hidden Brain for motivating me to write about this post today. They recently published an episode titled “Close enough: the lure of living through others ” and it resonates so much with me.
Do you ever find yourself going through video after video about a certain project you plan to do? Perhaps you are looking for instructions on how to do it, or simply looking for inspirations from others that have done it. It’s almost mesmerizing when we watch experts do what they are best at. And we somehow feel we can do it as well after we watch enough videos which is really an illusion. I find the same analogy also applies to our students: they watch us solve problems in class, and they may even find YouTube videos on the same topic and watch a few of those. And they tend to believe they can also solve similar problems after spending so much time watching. We all know how that turns out when we mark students’ test papers. They don’t know how, even though they’ve spent a lot of time watching others do it. Watching is not equal to doing. It’s a simple fact and yet many fall into the false belief that if we watch enough, we’ll become that expert in the videos.
I have to admit sometimes I make the same mistake: when I’m attending online courses, I watch others having discussions and feel I’m also part of them, even though I didn’t post a word; I feel I’m expert in the subject matter after browsing through what’s offered in the course, without actually spending much time on the listed learning objectives, only to find myself at loss when I come across the same problem somewhere else. In order to avoid this from happening, I tend to register way less courses nowadays, so I will have enough time to really sit down and study.
Last semester I had the chance to explore how would using Geo-Gebra in my Linear Algebra course affect student’s learning experience. My initial goal was to find out whether using it would improve student’s engagement in the class. I had taught this course for a few times by then, and one observation I made was how quiet the lectures were, compared with my other sections. In fact many students who took Linear Algebra last semester with me also took my other math course: Intro to Math Proof and we were discussing this interesting finding. They agreed that our Linear Algebra lectures were too quiet. They told me that Linear Algebra classes were not as engaging, and interesting as the other course even though my teaching style didn’t change. I had to do something.
I managed to secure a small funding from my university and did the following experiments. Once every two weeks, I will have four TAs going in to my lecture and sit among students. We will do one Geo-Gebra activity that helps students visualize and understand a new concept. For example, when we were learning the topic of eigen-value and eigen-vectors, students were asked to go here: Exploring Eigenvectors and Eigenvalues Visually and follow the steps:
Set the matrix M to be (1 0 ; 0 2)
Drag the point u until you see the vector u and Mu are on the same line. Record the value of lambda. How many times do you see u and Mu lying on the same line when u travel through the whole circle? Why?
Based on your observation, what can we say about the eigenvalue and eigenvector of M?
Set the matrix M to be (3 5; 1 -1) and repeat what you did above.
Check your lecture notes about the eigenvalues and eigenvectors of this matrix. Are the results consistent with what you observe?
TAs will be walking around the lecture hall and answer any questions students have. Some are about the applet itself, some are about the mathematics involved. By the end of the activity, a majority of students feel more comfortable about the two new concepts.
We did similar activities for a few other topics, and in general they helped students in understanding the abstract math ideas better. However, the interactions between students were not improved much. Most of them were working alone, and the class for most part was still pretty quiet. I’m in the middle of getting the data in place and measure whether the engagement level has improved or not, but my guess is there probably isn’t a significance improvement. Next semester if I’m to do these activities again, I will add at least one step: share your work with your neighbor and exchange what you have found with each other. And I will invite volunteers to talk about the questions, instead of me explaining them.
I will come back to this post once there are new findings.
Credit goes to the wonderful Angela Anderson . I have been following her amazing YouTube channel and weekly lessons and learned how to paint since two years ago. She’s a role model for me on my OER journey. She’s so generous of sharing her talent in painting and teaching and has reached such a wide community. Below are some of my recent paintings.
Research has shown student teaching evaluations are biased. Teaching evaluations evaluate gender, race, and attractiveness. Men are favored. White people are favored. To see a collection of resources that discuss the issues associated with teaching evaluations, see Rebecca J. Kreitzer’s post: EVIDENCE OF BIAS IN STANDARD EVALUATIONS OF TEACHING
If you are still interested in reading the full version of my recent teaching evaluations, they are here:
Disclaimer: Repeat Learner here refers to someone who can’t stop learning new things. I don’t believe the definition: a Repeat Learner is a student who has outstanding modules from previous years gives it justice. I’m a proud Repeat Learner who repeats the activity of learning new things all the time.
My life as a student lasted long. When I looked back, I spent almost twenty-two whole years (Y6-Y28) as a full-time student. At some point when I was near the end of my student days, I thought the learning part of my life was about to be done. How wrong was I! Once I started teaching as a full-time mathematics lecturer at Singapore Polytechnic, I quickly realized there were so many things I needed to learn: how to write a lesson plan that makes sense; how to communicate with students; how to write on the whiteboard/blackboard which minimizes the chance of anything getting erased during an one-hour lecture, how to navigate the LMS (we were using Blackboard back then), etc. I had a great officemate when I started my job, and she taught me new skills everyday in the first few months. I didn’t even know how to order textbooks! Once I settled on my new role as a lecturer and knew what I was doing, I found myself learning how to use Camtasia to make video lessons; taking online courses to learn about the newest edtech tools, and even a cool visualization software for statistics: Tableau. All these learning experiences keep my day exciting. They have brought much frustration and struggle, but also joy, satisfaction and fun. I’m in love with learning new things! It helps me master skills that make me a better teacher.
What I didn’t realize back then is learning new things can also help me stay humble and connected to my students. Sometimes I found myself quietly complaining things in my head while teaching: How can you not know this? How can you forget something that we just learned last week? How can you not get it? You see, I forget what it’s like to be a student, to be a learner who struggles. I took up painting two years ago, and whenever I can’t get things right, which happens to each one of my paintings, I tell myself this is what it’s like to be learning new things. Those quiet complaints in my head gradually go away. I’m able to put myself in my students’ shoes and see things in a different angle now. I’m more empathetic because I also struggle when I learn new things and I know that’s the good thing: without making mistakes and struggling, progress and growth won’t happen. I’m not suggesting every teacher to go out and learn something new today, but it’s important to remind ourselves what it’s like to be a beginner, a learner.
Now I’m challenging myself to learn how to play piano, which I figured might take years, especially after my first lesson. I’m not giving up just yet. The learning part is too good to walk away from. I guess I’ll never quit being a student.
I’m pleasantly surprised by how well-organized eCampus Ontario extend mOOC is, and hoping to make some meaningful and long-lasting connections with the community here.
As I’m going through the materials of module 2: Technologists, I started thinking of why digital literacy matters, and how much does it matter to “good teaching”.
Being digital literate starts with knowing the problems we want to solve. Is it about improving student’s engagement? Is it about deepening understanding of key concepts? Is it about communicating more effectively? Or is it about having a fun and open learning environment? While I was teaching in Singapore, every faculty member from my department took part in the popular Coursera online course: Powerful Tools for Teaching and Learning: Web 2.0 Tools.
That was the first time I took some time thinking about what exactly are digital literacies and why they are important. Our students today are different from students decades ago; in fact they are probably more digital literate than us in many aspects. In order for us to improve our teaching and reaching our students, being aware of what tools are available becomes very important. On the other hand most edtech tools have their own limitations. Before we use any new tool, it’s a good practice to try to understand what they are meant to help with, and what potential issues we may run into.
I believe most of the problems we are trying to solve have low-tech or no-tech solutions. Using digital tools is just one way of solving it. Perhaps before we dig in the endless list of “cool” tools that are out there, we should ask ourselves can we come up with a no-tech solution to address the issue at hand, focusing on the students, and what’s best for them.
I find the most useful way to adopt new tools effectively is by discussing it with colleagues and the bigger teaching community (we have a wonderful teacher community here: teacherforlearning channel! ). It comes to each one of us to share our experiences and spread it out, especially what don’t work. Don’t hesitate to share something that you tried and didn’t work. It might save someone else plenty of time and frustration in future. I stopped using Mentimeter a while ago because I didn’t like the paid version, and it’s hard to edit mathematics there.
For those who are taking the same course with me, you can find my extended activities of Module 1 here:
I joined the math department at University of Manitoba in 2020 Fall. Before that I taught math at University of Toronto Mississauga part-time since 2016 when my family moved to Canada from Singapore. I also taught part-time at two local colleges: Seneca College and Humber College while working at UTM. I taught math full-time at Singapore Polytechnic from 2012 to 2016. This is the space where I explore and share my journey of teaching mathematics, conducting education research projects, and learning about OER.