What do I need to get started?

You really just need three things to get started:

1) The course material and a way of keeping it organized.

You can download the course material from the Courses tab in the navigation bar. All the files to be downloaded are PDFs.

Once you have them, you need to keep them organized in whatever way suits you. Because the completed investigations are your "notes", and the course will frequently ask you to recall something you've learned in a prior investigation, organize them in a way that makes sense for you.

There are at least two ways to use/organize the investigations:

• You could be very “old school” and print them out and write on them with a pencil (don't use a pen for gosh sakes, see my discussion about making mistakes!). Then you can put them in some kind of old-fashioned notebook. That will all work, but that’s sooooo 20th century.

• Better, at least in my opinion, is that you download the documents to an electronic notebook (I have used Microsoft OneNote, but there are others, such as Obsidian, that are equally good, if not better.) Once in the electronic notebook simply “annotate” on the PDF. BUT…use a stylus. DO NOT try to type math. The math we’re learning here cannot be typed.

2) A graphing utility

We will do a lot graphing both in the investigations and in the projects. Once or twice I will ask you to graph something by hand, but most of the time you will use a graphing utility to do the dirty work. There are loads of free graphing utilities out there. I use Desmos, and I highly recommend it. But I have also used Geogebra. (If you are truly "old school", there is always a graphing calculator!) Whichever graphing utility you choose, make sure it has regression capabilities. We'll be doing a fair bit of that in the projects.

3) A willingness to learn math in a different way.

Learning by discovery is not always easy. Unlike traditional math learning, you won’t be spoon-fed with lectures and examples. Unlike many on-line math education resources, there are no videos, and there is no "gamification". Here you have to do the “work” of learning. (I put the word "work" in quotation marks because I believe that learning should be enjoyable or interesting, not "work". Maybe a better word here is "effort".

As part of this willingness to try a different approach, you mustn’t be afraid to make mistakes or to screw things up. Another reason to do the material electronically, not on paper. It's so easy to erase electronic mistakes and to try again! In my view, mistakes are essential to learning, something to be embraced, and to not be afraid of. These courses are all about thinking, trying, making mistakes, figuring out why, and trying again. That mindset is at the heart of any discovery/investigative approach to learning.

The Nuts and Bolts of Getting Started

Ok, you have navigated to one of the courses.

In each course you will find a syllabus and a set of modules. The syllabus is really just a glorified table of contents. You can scan it to see what lies ahead in the course, and you can use it to jump to any particular module. The heart of the course is contained in the modules.

Each module contains

• an introduction (to set up what you are about to learn)

• a set of investigations (to learn the new concepts of the module)

• a project or two that rely on the set of investigations

The process I've laid out is not difficult... just follow the sequence of modules. Start with Module 0, then move to Module 1, etc.

Once you are in any particular module, just follow the sequence there too. Start with the introduction. I think the introduction in each module is important to read because it states what you are about to learn and puts that in the context of the overall course. It also gives you some "food for thought" on how the material can be extended beyond the scope of the course.

After reading the Introduction, then move to the investigations...again, follow the sequence: Investigation 1, Investigation 2....etc.

Each investigation is step-by-step, guided questions or directions. Just answer the questions or follow the directions. By the end of the investigation, you will have, I hope, “discovered” the key concept that the investigation was trying to convey. As you complete an investigation, make sure you take the time to reflect that you have grasped the concepts of what you should have learned for that investigation (it’s detailed in the “Wrap Up” at the end of each investigation). This isn't about "pencil-whipping" through the investigations. There is no race here. Make sure you take whatever time is necessary to learn and understand the material.

Once you are done with the investigations in each module, attempt the project(s).

Once you are done with the investigations and the projects, you are done with the module....so move on to the next module.

And so on... Process-wise, it is really that simple.

But remember three things:

1) Your completed investigations and projects are your notes. So, write your responses clearly and completely. And keep them organized in whatever manner makes sense to you.

2) Don't look for "answer keys". These are not worksheets. There are a lot of right and wrong "answers" to steps because I ask you to explain things, in your own words. Don't be afraid to make mistakes; don't be worried about getting the "right answer" or the perfect answer. But do make sure you get the point of the investigation.

3) Don't get discouraged. Learning through investigations (and the projects) requires some effort on your part. First, you have to read the instructions in each step and then to do something. Graphing is an integral part of the learning here, so get ready to switch back and forth from your notebook to your graphing utility. Also, I don’t provide you with every last definition you need, nor do I try to explain everything or recall for you stuff you should have learned in prior classes. If you don’t remember something, or if a definition isn’t clear, then I expect you to go look it up (on the Internet or elsewhere) or to ask a friend. As I said previously, this isn’t some video where you press “Play” and watch somebody explain and work out examples. Here, you have to do the math.

Sequence, Pacing, and Grading

What about Sequence and Pace?

I have set the courses up so that the investigations in each module follow what I think is a logical order and the modules in each course follow a specific sequence. I suggest you follow the investigations and modules in the order presented.

That said, your brain may work differently from mine, or maybe you remember some of this stuff from before. If you feel you want to do things differently…go right ahead. The only thing I will warn you about is that sometimes an investigation will rely on one or more previous investigations, and a module may rely on one or more earlier modules.

As for pace, you set whatever pace you want. In a normal "school", the calculus course, for example, is a year-long course (two semesters). But in a normal school students are taking other courses along with this one. If you have the time, you can proceed more quickly. That said, take whatever time you need for the course. No one is judging you. The only thing I will suggest for pacing is that you start and finish an investigation in one sitting. It's awfully hard to pick up an investigation if you have only completed a part of it.

How are my investigations and projects “graded”?

There are no grades in this class. That's a good thing. This is not a "Paper Chase"; you are not doing this for the grade.

But I think it is important for you to “grade” your own stuff. Indeed, your self-grading (in edu-speak this is called "reflection") is an important part of the process.

For the investigations, it is relatively simple to check your work. For most, if not all, the investigations, I ask you graph this or that. The graphing is an important check, not a throw-away.

For projects, there are plenty of possible “wrong” answers, but there are no “right” answers. Again, the graphing I ask you to do is part of the reflection process to uncover “wrong” answers and correct mistakes.

If you are working with others, you can have them “grade” your investigations and projects. Collaborative learning, done properly, can be excellent.

Other Questions

“Can I use outside resources to help me?”

Absolutely. In fact, I encourage it.

Remember, the focus here is on understanding and applying; the focus is NOT on skills or memorization. (In my view, anything that can be answered by technology is very “low level” of learning and, bluntly, not all that interesting.) Therefore, the use of technology does not undermine the goal of the course. A good graphing utility is essential for this course and as a means of checking your work. As I said earlier, I use Desmos, but there are plenty out there.

Moreover, I fully expect that students will encounter things during the investigation that they may have forgotten from previous math classes. Go look it up! If the algebra is too difficult and impeding your progress toward understanding the concept, use an app. It’s ok. The bottom line is that it does not matter if you have forgotten something. The important thing is that you take the initiative to know what you need to know and go find that somewhere. Isn’t that how life generally works?

“Can I work with others?”

Yes and no.

I encourage you to work with others on the investigations. The “answers” there are the same for every student. But, do not simply let another person tell you what to put in response to a step. (That's one reason I am not putting up "Answer keys" to the investigations. Understand what is being asked and how it should be/can be answered. A person telling you the answer (without giving you an explanation you understand) is not helping you.

As for the projects…you should be “own your own”. The projects are designed for you to assess your understanding of the material in the module. Your project should be different from anybody else’s.

“Do I need to study before I do a project?”

Heck no. Despite the fact that I was a good student, I am not a fan of studying. Too stressful, and I hope to eliminate (or at least reduce) the stress of learning math. Also, studying implies there are things you need to memorize. As I used to tell my students, memorization is for those things you don’t want to think about. For example, basic multiplication tables should be memorized. But do you really need to memorize the “rule” for the derivative of sine function? After using it enough times, you probably will memorize it, but that’s a different thing.

If you are in the middle of a project or an investigation and you need to remember something from one of the investigations in the module (or one of the investigations in a previous module or even something from a "lower level" math class") and you have forgotten it… go look it up. It's OK! The investigations are akin to your “notes”. Keep them organized, know where and what to look for. Then as you approach a project, think of the project as being “open-notes”.

I'm in the middle of an investigation and I get stuck. What do I do?

Sometimes you do get stuck and have to figure a way out. Try something. Try anything. Put things down for a minute. Analyze what you don't understand.

It's ok. It happens.

In education terms, it's called "productive struggle". In real-life, it's called learning how to problem solve.

In some of the investigations I have put in "HINTS" where, based on my experience using these investigations in the classroom some students have struggled.

If you are working with someone else, ask them. If everything has failed and you're completely stuck, send me an email. (But remember, I am retired and not on email continuously.)