*(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 . 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.*(I’m sure many people have cringed again)*

** 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?

These are three very good points. CBL and PBL are very powerful if the instructor is well prepared. I definitely think the second point you raised would pose a more significant challenge for planning a transdisciplinary class, and was wondering if you have any suggestion on that.

Transdisciplinary classes definitely pose a few more challenges. That being said, it will require a little more planning when it comes to providing adequate background and context. I definitely prefer holding traditional lectures before beginning any CBL. Despite this helping most students catch up to speed, some students will definitely be further behind others in the assumed prerequisite knowledge (as is the case with most transdisciplinary classes). Unfortunately, you have to decide after a certain point how much class time you are willing to dedicate to background knowledge in a way that won’t hold back the students proficient in the case/project topic. At that point, I would suggest offering supplemental materials for the students to explore on their own or offering separate out of class time to provide missing prerequisite knowledge.

Wow, this was a long but very well thought out post!

Looking back on my math classes, I only had one class that used case-based learning. At the time, all of us students were really confused on why we suddenly had to do a project instead of the usual problem sets. Afterwards, I realized that it was actually quite helpful in illustrating the usefulness of the concepts we were learning, and way more engaging.

Do you find that your students are fairly open to case-based learning, or confused by the novelty of it like my classmates and I were?

The times that I’ve implemented it and the times I’ve seen it implemented, I find students are typically confused by the novelty of it. You’d expect something like it in a science class, but not usually in your math classes. This is understandable though because it takes much longer to prepare for than going through the same lectures and examples you’ve done for the last however many years you’ve been teaching. Despite the initial confusion, students usually come around to the disruption of the same-old same-old they’ve gone through for past math classes.

Thanks for sharing! I appreciate your well-thought-out points and the specific examples that you provide. I agree that CBL could play an important role in retention. Like we heard in class the other week, it’s sometimes easier (and arguably more meaningful) for students to recall a case than an exact principle. I remember one CBL-style math class in undergrad (differential equations). One of the things that made the class effective from my point of view was the group work and the interaction with the instructor and the TA’s during class. I could tell the instructor and the TA’s spent time preparing (to your first point) and it made the world of a difference when asking questions during class.

That sounds like a great experience for a higher level math class. Do you mind me asking what the group work was exactly? My experience in differential equations was adequate, but I know it had to be boring, dreadful, etc for the other students since it was purely traditional lecture. I’d love to find a more involved way to teach differential equations.

Additional question, which is not as important, what type of differential equation class was it? Ordinary, partial, etc.?