OCMA 2019: Improve student engagement using Geogebra

I presented my talk at The Ontario Colleges Mathematics Association
39th Annual Conference, on May 23rd. The talk can be found here:

I shared what I did in my Linear Algebra class: using GeoGebra to explain and visualize a few core math concepts including linear systems, complex numbers, and eigenvalue/eigenvectors. The audience worked together and created this Padlet during my talk.

I also shared the challenges I encountered while doing this education research project. It’s great to have a conversion with colleagues from all different institutions and learn from them. I appreciate the opportunity and hope to be there again next year.

[2019SMAT102] Videos for AM-GM inequality, inequality and set related proofs

I made several videos explaining a few topics that we cover in this course. Some of them came up during my office hours. If you’d like me to explain a certain concept/question covered in our course, feel free to drop me an email and you might see it here 🙂

  1. AM-GM inequality: the statement, how to prove it and its application.

2. Given x > y >z, prove xy+yz > \frac{(x+y)(y+z)}{2}.

3. If  A \cup B \subseteq C \cup D, A \cap B = \Phi, C \subseteq A , then  B \subseteq D.

[BAB210] Business Statistics

I’m teaching this course during the summer semester at Seneca College. The course outline can be found here:


We are into our second half of the semester now and being able to use Excel is an important part for the successful completion of it. I made several short videos explaining how to use certain built-in functions in Excel. More will be coming up as our course progresses.

  1. How to draw a bar chart for categorical data:

2. How to build frequency distribution table and draw histogram:

3. How to do cross tabulation:

4. How to use descriptive states tool in Excel:

5. Excel functions to find mean, median, mode,variance and standard deviation:

6. How to find covariance and correlation for two sets of data:

7. How to draw a box plot:

8. How to draw comparative box plot:

9. Continuous Probability Distributions part I:

10. Continuous Probability Distribution II:

11. how to select random samples

12. How to construct confidence interval for a population mean with population  \sigma known:

13. How to construct confidence interval for a population mean with population  \sigma unknown:

14. How to construct confidence interval for a population proportion:

15. Hypothesis Testing of population mean with  \sigma known: H_0: \mu=295, H_a: \mu \neq 295: two-tail test

16. Hypothesis Testing of population mean with  \sigma unknown: H_0: \mu \leq 7, H_a: \mu > 7: upper tail test

17. Hypothesis Testing of population proportion: H_0: p \leq 0.20, H_a: p 0.2 7: upper tail test

18. Regression Analysis:

[2019SMAT102] A letter to my students

Dear all,
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.

Have a great semester!

MAT223 Linear Algebra Exam Review questions answered in videos

Dear students from MAT223:

I’ve received some requests to explain the following questions and I made a few short videos to explain how to solve them.

Happy learning!

  1. If $T$ and $S$ are linear transformations $ \mathbb R^3 \rightarrow \mathbb R^3 $, and $\displaystyle \vec{u}, \vec{v}, \vec{w}$ are three distinct vectors in $\mathbb R^3$ so that

 T(\vec{u})= S(\vec{u}), T(\vec{v})= S(\vec{v}), T(\vec{w})= S(\vec{w}),

then $\displaystyle T(\vec{x}) =S(\vec{x}) $ for all $ \displaystyle \vec{x} \in \mathbb R^3.$

2. If A is 3 \times 3 matrix and dim(null(A-I))=3, then A is diagonalizable.

3. Consider

     \begin{equation*}A=\left[ \begin{array}{rrr}2 & 4 & -1  \\0 & 7 & 2  \\0 & -2 & 3\end{array} \right]\end{equation*}

Decide whether A is diagonalizable. Justify your answer.

4. What is the matrix of a linear transformation

 T : \mathbb R^4 \rightarrow \mathbb R^4


    \begin{equation*} T(\vec{e_1})  =\vec{e_2} = 3\vec{e_3}; \\ T(\vec{e_2}) =2 \vec{e_1} - 3\vec{e_2} + \vec{e_3};\\ T(\vec{e_3})  =3 \vec{e_3};\\ T(\vec{e_4}) =-\vec{e_1} - 4\vec{e_4} \end{equation*}

Compute the rank of this matrix.

Are you living vicariously?

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.

Reflection of using Geo-Gebra app in Linear Algebra course

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:

  1. Set the matrix M to be (1 0 ; 0 2)
  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?
  3. Based on your observation, what can we say about the eigenvalue and eigenvector of M?
  4. Set the matrix M to be (3 5; 1 -1) and repeat what you did above.
  5. 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.

My Acrylic Paintings

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.

2019 Winter MAT102: Intro to Math Proof

The main topics that we cover in this course are:

  1. Numbers, and Inequalities.
  2. Sets, functions and fields.
  3. Informal logic.
  4. Mathematical Induction.
  5. Bijections and cardinality.
  6. Integers and divisibility.
  7. Relations.

The course note is available: MAT102

The slides for weekly lectures can be found here:

  1. MAT102_Week1-2
  2. MAT102_Week2-3
  3. MAT102_Week 4-5
  4. MAT102_Week 6
  5. MAT102_Week 7
  6. MAT102_Week 8
  7. MAT102_Week 9
  8. MAT102_Week10
  9. MAT102_Week 11-12

Below are a few short video clips I made to explain certain questions upon students’ request.

  1. 2.5.29 Part a) Prove or disprove if two functions f and g are bounded, then their sum f+g is bounded.
  2. 2.5.29 Part b). Prove of disprove if two functions f and g are bounded, then f^2-g^2 is bounded.
  3. 2.5.50. Show that in any field F, the equation x^2 = 1 can have at most two solutions.
    Can you think of a field in which the equation x^2 = 1 has exactly one solution?

CV, Teaching dossier and more

Here’s a copy of my recent CV (edited in May 2022)

Here’s a copy of my teaching dossier (edited in Dec 2021)

Here are a few gems I collected during my journey as a math educator: letter1, letter2, letter3, letter4, letter5, letter6, letter7, letter8, letter9, letter10, letter11, letter12

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:

MATH1500 Calculus I (2021 Spring)

MATH2150 Multivariable Calculus (2021 Spring)

MATH2720 Multivariable Calculus (2021 Spring)

MATH1510 Applied Calculus I (2020 Fall)

MATH2720 Multivariable Calculus (2020 Fall)

MATH2030 Combinatorics (2020 Fall)

MAT202 Discrete Mathematics (2020 Spring)

MAT135 Calculus I (2020 Spring)

MAT135 Calculus

MAT102 Intro to Math Proof

MAT102 Intro to Math Proof

MAT223 Linear Algebra