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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I often leave replies!


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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.

Tips on Collaborating Better

As a student in an interdisciplinary program, learning to collaborate effectively is one of the most useful skills I can have. Although it may not be as important for other disciplines, it is still a very useful skill and can help in more ways than research. Collaboration relies heavily on communication (as you will see), but here are a few tips I’ve learned that are extremely important to collaborating effectively:

  1. Know your audience.
    Knowing your audience is one of the simplest things you can do when collaborating with someone. When you know your collaborator’s familiarity with a topic, it is a lot easier to adjust the level of complexity in your explanations and questions. By doing so, you can avoid jargon speech or miscommunications. The best way to get to know your audience, is to simply ask. For instance, you can ask if your collaborator is familiar with a term you use frequently to gauge their knowledge of a core subject. Based on the response, you can continue your question or explanation in a way you would with a colleague or pose your explanations or questions in a manner you would to new students entirely unfamiliar with a concept… Just be careful not to be condescending in your simplifications!
  2. Know yourself.
    Before you can begin to collaborate with others, you need to know exactly what you need from others. Make sure you know exactly what skills you are lacking, so you can be sure you are collaborating with someone who is proficient in those skills. This will also help you guide your questions and expectations for your collaborator. Even if the first person you contact for collaboration does not have the skills you are looking for, by having a concise explanation of what you need, this person could potentially put you in contact with someone that does have the skills you need.
  3. Keep an open dialogue.
    This is more of a communication tip that should be utilized with all individuals you work with. You should make sure that you keep an open dialogue of expectations and desired outcomes. You should make sure that issues such as authorship and deadlines be hashed out and agreed upon early on in a collaboration. More importantly, keeping in touch with current progress and goals can help make sure everyone is on board and achieve overall goals.

What are some tips you’ve learned in your own experiences with collaborations? Comment more tips below!

Math pun: Why is 1/5 always so stressed out? Because he’s two tenths (too tense)!
(Today’s Math Pun brought to you by algebra)