Good Learners Make Good Teachers

It’s hard to say what exactly will help you find your teaching voice and your identity as an instructor, but I can certainly say, from my experience, being a good learner is imperative. I found that one of the most important things to finding my teaching voice was finding what was important to my own learning experiences. In order to do that, I had to be an active learner in any experience I could. What was it about a particular class that made assignments manageable to complete? What were the most important things instructors did that made lectures consumable?

Being able to reflect on all of these things and culminate the best experiences (and the lessons from the worst) into your personal style and firm teaching ideals is all that is needed to find your way to a teaching voice. Personally, I feel that as long as I am keeping a good sense of empathy for my students and fostering some sense of critical thinking or academic exploration, then I am successful as an instructor. This is not to say that I will find a set of ideals and stick to them forever; things change and so do we. It’s important to not just stick to a particular set of rules forever. Obviously, the important things like empathy, integrity, and critical thinking should be the foundation for all teaching philosophies, but that doesn’t mean how we express our teaching voice must remain the same.

Times change, students change, needs change, and so must our teaching voices. In order to do this, not only do we have to be good learners, but we have to be constantly learning. Being a good teacher isn’t an irrevocable status, the definition of a good teacher changes with the needs of students. So, as teachers, our methods and values must change and, consequently, so will our teaching voices.

Math Joke of the Day: What do you get if you divide a pumpkin’s circumference by its diameter? Pumpkin π

The Importance of Framing

Just about all of us have been there: that one dreaded math class that feels like a drag on your whole day. For some people, that’s been just about every math class. And for others, we’ve almost never been there (myself included). This typically stark contrast in outlook on math makes me wonder at what point does math education becomes less fun and surprising and more confusing and mind numbing?

Our recent discussions on critical pedagogy, made me realize that the biggest issue is framing. A lot of math classes take on the same form: topic introduction, repetitive practice, testing/assessment. Although most math classes take on this form, some are more bearable than others. So, what is the difference? It must be the framing. Here, by framing, I refer to how material is presented and the attitude used to approach it. For some people, the idea of learning a method and finding different ways to repetitively use it is mentally stimulating enough to stay interested. I find myself in this group of people; it’s like learning a new rule in a puzzle game and you get to play puzzles all day. However, for others, this is a nightmare. This kind of framing hearkens back to the need to educate students to repetitively carry out a task to be successful in assembly lines which were integral to the work force in the 1920’s.

With the rise of technology, this sort of student is no longer needed and repetition no longer provides students with the skills they need. Consequently, our framing of education, in general, needs to change as well. We no longer need that repetitive driving of knowledge into students’ minds. Instead, we should be constantly finding ways to incorporate different teamwork, critical thinking, and diverse problem solving skills. Our framing of education should be done in a way that can still excite students, whether it be game-style lectures, problem/project-based learning, or simply reworking/rewriting traditional lectures to include relevant topics to students. Of course, this is easier said than done, but I believe simply making an effort will make a world of difference. So drop that textbook you’ve been teaching from for the past however many years, and find a new way for students to discover information instead of trying to cram it in their heads.

At what point did the framing of your math classes become boring or unbearable for you (if it ever did)? If it didn’t, what kept the fun going for you?

Math Joke of the day: You might think my calculus jokes are derivative, but they are an integral part of me.

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Learning Math in a Digital Space

(long, but consumable post incoming!)

Learning math in a purely digital space is a pretty unique experience that comes with a lot of advantages, but A LOT of pitfalls. It is wonderful to see that online and distance learning has made education more accessible to so many people. This also means that it is also starting to change the shape of learning environments. Here are some do’s and don’ts I’ve picked up from learning math (along with a few other subjects) in a digital space.

Online Math Do’s:

  • Try making time for recorded live lectures. Choose a lecture time in your day that works for you. This will make each of these lectures feel unique and tailored to the needs of the students in each offering of the course. This, of course, can be a bit demanding, so perhaps choose a few lectures out of the course (preferably the most important or tricky ones) that you will present live to your students, so you may answer questions immediately and save yourself (and your inbox) a lot of trouble.
  • Record your lectures. Period. (This one applies to any subject!) Regardless of how you decide to present the information to your students, give it a little human touch. Nothing is worse than having to scroll through tons of wordy powerpoints in silence. The online/hybrid class should feel like a class, not like a required reading in a textbook. You want to make this experience feel as close to a traditional class as possible. This will also give your students the ability to change the pace of the lecture as needed. I can not tell you how many times I had to change a lecture’s playback speed to 1.5x because the pacing was too slow. Contrarily, students also get the advantage to pause and linger on pieces of a lecture that were particularly challenging to understand.
  • Use this technology to your advantage! One of the best things about teaching math in a digital space is that the technology affords you so many advantages that you can not always implement in a traditional classroom. Being able to spend time looking at animations and graphics in a lecture were always a mental treat. In traditional math classes, hand drawn graphs and concepts could be a little tricky for students to understand and copy (depending the instructor’s artistic abilities). Additionally, very few instructors use powerpoints and other digital means of presenting material in traditional setting, so pulling out a laptop and setting up a projector could often detract from the learning. These are no longer problems in the digital space, so generate some neat graphics and animations! They’ll flow seamlessly into your lectures and students will definitely appreciate the clarity in graphs!

Online Math Don’ts:

  • DON’T use online assessment tools (where possible). This tip boasts the same sentiment as last week’s post. Do your best to avoid tools that automatically assess your student’s assignments. Despite all the good they could do, they tend to do more harm than good while students struggle to get the computer to acknowledge simple human errors. Clearly students should receive no credit for incredibly small things like missing a negative or answering with two decimal places instead of three. Small clerical errors are not a sign of a lack of understanding, but a sign of simply being human. If it is both financially and physically feasible for you as an instructor (hopefully with access with teaching assistants), then you should NOT use online tools to assess work. The only exception to this should be for extremely large class sizes… which at that point you should already know there is not going to an easy way to complete grading without complications.
  • Don’t try to dictate your students’ schedules. Give suggestions and light warnings instead. We have all heard the warning “you can not do this in one night”. Yet, somehow, just about all of us have miraculously done it in one night. It is fair to try to warn your students of the effort an assignment may require, but understand that many students will try to test the bounds of required effort anyway to their own detriment. My biggest example is that of an online secondary education course I took in college. The instructor proudly stated that we should not attempt to finish each week’s assignments in one day because the course required about nine hours of study and work weekly. Of course, my stubborn response was “Nine hours? That is one day”. So, I started and completed each week’s assignments in one day. This drastically changed my expectations for the course and I did not take it seriously. I passed the course with good grades, but I retained very little detailed information from it because all of the learning was crammed into very boring nine hour blocks of my Sundays. Moral of the story: For your students’ benefit, give them suggestions on how they should prepare for your class, don’t give them a time frame they will obviously cram with little to no material retention.
  • Don’t make assignments closed book. This one is a tricky one, but let’s be entirely honest here: they are going to look at their notes anyway. We would like to think that our students will follow the honor code to a T, but the reality is that they won’t. Clearly we want our students to demonstrate their learning without the training wheels of their notes, but teaching in a digital space makes it a little difficult to enforce. Here are two solutions to this problem:
      1. Allow open book assignments, but change the available time frame for quizzes and tests. You will want the time frame to be suitable for students to glance at notes to get necessary formulas, theorems, and concepts, but not long enough to entirely familiarize themselves with those topics as if they haven’t studied. Bottom line, you want a ‘glance’-able time, not a ‘study’-able time. This should be somewhere around 5-10 minutes longer than a normal closed book time frame since quickly going back and forth through notes will take away some time.
      2. Do not allow open book assignments, but require summative assessments be taken in person. Since many students of an online course will be near your institution anyway, set a time and physical place to take tests and exams. This will force students to demonstrate their understanding while limiting open book cheating. This, of course, comes with many limitations. Not all of your students will will be able to meet at a single time. In this case, work with individual students to take tests/exams at a different time. You will, of course, want to limit this to prevent some students having an advantage over others. It may be wise, to gauge the general availability of your students before setting a time. For distance learners, the main institution’s campus may be feasibly inaccessible. In this case, you may want to coordinate with a test-taking facility (or instructors from a closer institution) to have your distance learners take proctored tests/exams and have their results scanned and emailed or faxed to you. (See? Fax machines can still be useful. They aren’t the relics we make them to be)

What are some other do’s and don’ts your instructors have (or you wish they had) implemented in your online/hybrid math classes?

Math Pun of the day: What do you call a young eigensheep? A lamb, duh!

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Digital Pedagogy in Mathematics

It is a very difficult thing to find the right blend of tech and tools in the mathematics classroom. On one hand, you want to have as many educational tools available for your students to aid in learning. On the other hand, it is difficult to determine which tools to which your students will respond well. I’ve seen many tools implemented well and others not so well.

When it comes to mathematics there are some tools that just make sense in the classroom. You certainly wouldn’t want to prohibit calculators or touchscreen tablets/laptops. (Personally, I would like to ban calculators in the mathematics classroom, but the problems with that are a another blog altogether!) These sorts of tools just go hand in hand with making learning easier for students. Additionally, it is fairly easy to gauge how distracted your students are with these tools. You can expect students to spend no more than a few seconds on a calculator and it is extremely difficult to take math notes if a touch screen device is used for anything but a paper substitution.

Unfortunately, there are some common tools that I think do more harm than good. The biggest perpetrator in my book are online mathematics assessment tools (i.e. online homework, quizzes, or tests graded by the computer). In my years tutoring and teaching math, I have never had more problems with anything like the online assessment tools! These sorts of tools are supposed to easy tools that allow students to have multiple practice problems and guided example solutions as well as saving instructors time with grading formative assessments. The tool is great in theory, but the implementation… Oh, the implementation.

In math, there are just some things a computer can not do. Just like the humanities, a computer can not assess a student’s true understanding of a topic. Something as simple as a misplaced negative sign can result in a wildly different answer from the correct one. Many of these assessment tools can not identify these small problems and adjust partial credit accordingly. More often than not, students simply get no credit for the problem and their resulting grades are not representative of their actual understanding. Some people may think, “This is fine. Students need to be diligent in their work and be mindful of mistakes”, but the problem gets much worse. Many of these assessments are very picky with the form of their answers. Students will often lose points for rounding to 3 places instead of 4 or for not simplifying the exact way it is looking despite equivalence in answers. There is nothing more frustrating than fighting with a computer over what you know is right.

There are so many tools I support in the mathematics classroom, but online assessment tools are BANNED. I do not mind having to go through all assessments by hand if it means my students get the feedback they deserve. If time is a factor, say there are too many students, then it is time to adjust how I’m evaluating my students (limiting the number of assignments, reducing the length of assignments, getting more teaching assistants, etc). I feel this is more fair to my students and aids in learning a lot better. I’m open to just about any tech or tools in the classrooms except online assessment tools until artificial intelligence advances enough to not penalize human error so heavily!

Math Pun of The day: What is a frog’s favorite function? The deribbitive.

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Going Beyond the Textbook in Mathematics

(This one is a long one this week. Be prepared!)

Mathematics has always been one of those topics that has the power to make many people cringe when they think of their studies. Many people will have flashbacks to word problems that made no sense, that unit that just never seemed right no matter how many times someone explained it, and countless formulas to memorize. It’s just one of those course subjects that when done wrong can leave a long and lasting impact on students for the worse. Fortunately, when done right, it can also leave a positive impact on problem solving and critical thinking even if mathematics does not play a large role in the student’s future. However, that’s the problem: How in the world are you supposed to do it right as an educator?

I do not claim to have an absolute answer to this question. There’s no one way to reach all students when it comes to mathematics. But, I believe I know what can help us come closer to a solution for this problem: Case/Problem/Project Based Learning (CBL).

This week we’ve discussed the sorts of impacts CBL can have on students. They can certainly improve a student’s understanding of a topic, develop marketable skills, and, in some cases, provide future career networks. With these benefits in mind, why would you not want to incorporate CBL into your classroom? Despite its wonderful advantages, it has some drawbacks. There is quite a lot of planning and resources that go into effectively incorporating CBL. For instance, does your project require new programs that you’ll have to acquire licenses for? Do all of your students have access to adequate computers and internet to be successful in solving their case, project, etc.? Do your students have the background knowledge and skills to be successful from the start? Do you have the equipment necessary to make the observations required for your case, project, etc.?

With this in mind, here are three ways I suggest CBL be incorporated in the classroom in the context of mathematics.

1. Plan Ahead

Just like any lecture, review, or teaching experience you’re in charge of, you should always plan ahead and plan thoroughly. CBL is tricky to incorporate in a class, let alone a mathematics class. As a mathematics educator, especially in higher level mathematics, it is very easy to see how quickly example problems become large scale. Take, for instance, calculus (I’m sure many people have cringed again). When teaching students calculating rates of change, most example problems aren’t small scale like the amount milk leaking from a glass; instead we usually see examples focusing on important structures like water or oil tanks. Getting the measurements from a leaking oil tank isn’t something you can easily do (from start to finish) in an afternoon. These sorts of things likely require a partnership with a township or company, which will in turn require lengthy communication on access to facilities, data collection, and various legal permissions. Even if you decide to reduce the scale of a problem, you have to decide how you are going to change the scale, how and where you are going to get the materials to collect your data, and how you intend this smaller scale problem to help your students gain real world experience. Even low resource solutions to incorporating CBL, like the case studies that can be found in the National Center For Case Study Teaching in Science, require some thought and review on your part as an educator to determine what sorts of skills and resources your students need to be successful.

Bottom line: Be sure to plan way in advance for any project, problem, or case you intend for your students. Put in the time; your students deserve it.

2. Provide Background and Context

There is nothing worse than being given a project and not knowing where to start. The point of CBL is to develop a deeper understanding of an area, but if there is no foundation to build on, it is very unlikely your CBL implementation will be effective. Last week, I promoted a balance between traditional lectures and relevant applications like CBL. Although, it is not required (no matter how much I advocate it), it is extremely imperative that you confirm that your students have all of the knowledge and skills necessary for success. Your students are not going to spontaneously grasp a concept by struggling with it alone. (Of course, there’s an extremely unlikely probability that this could happen, but why would you put your students at such a disadvantage?) It is entirely up to the educator’s discretion on how background knowledge and context are provided to the students, but it needs to be provided. The direction and aims of the CBL could potentially be missed by the students if they are uncertain of the background and context. You would leave the students open to gaining the wrong experiences from the CBL. Additionally, you could also save your students time that would be spent trying to research the background skills and context that could be spent on completing their project. Instead of leaving your students to try to learn and gain a conceptual understanding of derivatives, teach them how derivatives are generally used and applied in the real world (or provide them with the resources to do so on their own).

Bottom line: Set your students up for success and save them some time by dedicating time to traditionally teaching them background knowledge or providing them with resources so they may obtain it on their own.

3. Make it Matter

Arguably the hardest part of math classes is making any of the course topics matter. Despite how useful various mathematics skills are in a wide array of areas, it is difficult to pique a students interest in calculating the width of a lake they will never be in or determining the arrival of a train when the train company already has an app that will tell you its ETA. Granted, there is a lot more work in finding and modifying examples that mean something to your students, but that is part of planning ahead. The point of CBL is to give your students a unique and memorable experience with a topic in the hopes that it will strengthen their understanding. If their hearts aren’t in the topics they’re working on, then you will likely receive low effort work and the students will likely dread the whole experience. Try gauging your students’ interest in a particular CBL topic or, if you are certain your students will be indecisive (as a whole) about topic choice, consider allowing them to choose their own topic within an area. Allowing students to choose their own topics may be a bit of a hassle when ensuring their background knowledge, but you will definitely receive heartfelt work. Whatever you decide to do as an educator, make sure the experience is fun and worthwhile for your students.

Bottom line: Choose your CBL topic wisely. Make sure you are focusing your students’ efforts on something that is going to matter to them as well as help them.


Of course, these tips can be easily generalized for other topics, but CBL can play an integral role in mathematics. For students whose career path or academic major require more mathematics courses, it is crucial that your students retain the course information. CBL could be all that is needed to make a lasting impression of a topic to aid in retention. A simple change in humdrum of most mathematics classes could help us fix this natural cringe many people have when mathematics is mentioned.

What do you think? What was the worst part of math classes for you? How do you think your educators could have done better or how CBL could have changed the experience? Did I miss anything?

Math Pun Joke of the day: Why did the chicken Cross the mobius Strip? TO get to the same side.

Balanced Learning

With the rise of diversity and inclusion in higher education, the needs of students have drastically changed. The same methods that have been used for decades may not carry the same efficacy as it did before. A push for more aberrant methods may be exactly what higher education needs, to an extent. Although I fully support more creative and unique methods of teaching, I do feel swinging too far to the other side of the “teaching spectrum” so-to-speak could be harmful.

The most impactful higher educational experience I’ve had came from a course called Infectious Disease Epidemiology. Aside from the collectively taught curriculum, the thing that stood out to me the most was the consistent balanced structure of the course. In each unit of the course, professors would combine traditional lectures with some case-based learning project at the end. As someone who thrives in a traditional lecture setting, it made me uneasy yet confident in the work I was doing. Although I struggle with the critical thinking and ingenuity typically required of case-based learning projects, I was allowed to think about topics pertinent to the case objectively and gain an arsenal of knowledgeable tools beforehand.

In this course in particular, I was allowed to learn how others had used important strategies and protocol before personally applying them to a similar case or situation. Students were lectured on important topics before given their case to work on. Contrarily, I am sure working through a case together from start to finish develops a more natural understanding of a particular problem or topic. However, I feel this combination of traditional and contemporary methods can provide a better education for a wider range of students. Seeing as not every student has had their critical thinking nurtured appropriately by their academic settings, this combination of methods could prove optimal.

Overall, I feel that case based learning can be an incredible thing to include in the structure of a class. However, its implementation should be done with caution as to not exclude or create any difficult boundaries for students that may not have developed the necessary critical thinking skills to hit the ground running in a case based learning scenario. That is, a healthy balance between traditional lectures and case based learning projects could enrich learning for all students.

Math Pun of the day:  Any shape is a circle if you treat the radius as a variable!

Getting Personal: Diversity and Inclusion

Hi all,

Today, I wanted to emphasize the importance of diversity and inclusion especially in the sciences. It is so easy to feel overwhelmed as a woman in the sciences, let alone a black woman. Have you ever walked into a room and realized you were the only person of your gender identity? Your race or ethnicity? Your background? Your personal views? Your religion? I certainly, have and, let me tell you, it was frightening. It can be so incredibly difficult to feel comfortable enough to be yourself and reach your full potential when you are in an environment that doesn’t seem all that welcoming.

I have had a wide range of experiences in different labs and academic settings that have taught me so much about thriving in environments where diversity and/or inclusion may be lacking. Here are a few things that I’ve learned that can help you survive in places that don’t seem all that diverse or inclusive and how to deal with the cards you’ve been dealt.

1. Check yourself.

Remember my question about walking into a room and feeling “other” in some way? How much did you really look before you made this decision? Oftentimes, we make these sorts of decisions before we actually get to know everyone in the room. When we talk about diversity and inclusion, we know that we should never judge a book by its cover. That is, we know that we should never base our final judgments on  a person solely on what we can immediately see. So why do we do that when we decide that we are “other” in a room? Perhaps you may not share the same ethnicity or gender with someone in a room, but you may share the same sexual orientation, background, religion, or interests instead. Be careful to not make assumptions about others, you might be surprised what lies underneath! I have met some wonderful and interesting people under the most unlikely appearances. Make sure you get a chance to know everyone before you decide you’re alone! Although we may often base our main perception of diversity on what we can see, diversity is so much more. So, don’t exclude yourself before anyone gets a chance to include you!

2. Be Brave!

As I said before, when we’ve already decided we are “other”, the situation can be absolutely frightening. And when that’s the case, we can often feel discouraged from reaching out and getting to know people or allowing ourselves to be comfortable. The only solution to this is to be brave. You can never fully assess the diversity or inclusion of a given environment until you give others a chance to show their true colors. You may not like these colors all the time, but there are definitely people out there that make it worth the risk!


3. Fight or Flight?

Not every situation or environment is going to be welcoming. It is not ideal, but it happens. Perhaps your environment has more people that rub you the wrong way than you prefer. In this situation you have two options: fight or flight.

Fight: There is more than one way to fight a lack of diversity or inclusion in an environment. The most obvious way is constructive and respectful confrontation. That is, directly address the people or policies that may be affecting diversity and inclusion. However, confrontation isn’t for everyone, myself included. For those that can not handle confrontation, you can still fight by doing your best to make others feel welcome and spending more time with the people more willing to contribute to a more diverse and inclusive setting.

Flight: Although I fully support people fighting for diversity and inclusion in their environments, after a certain point you have to re-evaluate the worth of your time and concern. It is hard to say, but some environments are a lost cause or require more fighting power than you can provide on your own. Know when to cut your losses and recognize when a setting is no longer good for your mental and social health.


I hope these tips can help you all feel more comfortable in or learn when to leave environments that are lacking in diversity or inclusion. They have certainly helped me!

Math Pun of the day: If 666 is an evil number, then 25.8069758011 is the root of all evil