How is Discover Math different from traditional math instruction?

Traditional math teaching follows the “I do; we do; you do” methodology.

The teacher provides a lecture on the skill (or concept) to the learned. And the student sits there and listens and takes notes (sometimes).

The teacher works through an example (the “I do”) of how to do the skill. And the student sits there and tries to follow along (maybe).

The teacher gives out worksheets (or some variation thereof) for students to practice in class (the “we do”). And the students attempt do the set of problems on the worksheet.

The teacher assigns homework for the student to practice on their own (the “you do”). And the "good" students do the homework. The rest of us let it go, unless it's graded.

After completing this method for a number of skills or topics, the teacher assesses the students’ learning through quizzes and/or tests.

Most on-line instruction on math follows this same methodology (leaving out the "we do" part of things for obvious reasons).

The Investigation Approach used here turns traditional learning on its head.

There is no "teacher" here or any videos to watch. You don't copy the teacher's notes and examples.

Rather, you learn by doing investigations. These investigations have very little or no lecture up front to “teach” the concept or demonstrate the skill. (I do provide some definitions where necessary.) You uncover the concept to be learned through a series of steps in the investigation.

There are no worked examples or worksheets or homework in Discover Math. Examples and worksheets are good, maybe, at building "skills". But that's not what this course is about. Instead of focusing on skills or processes, the investigations focus on how those concepts are applied or used.

There are no tests or quizzes here. Tests and quizzes are fine for assessing procedural skills, but are not very good at assessing comprehension of concepts. In Discover Math, you can assess your own progress through projects, which are designed to get students to think deeply about whatever topic we're learning in that module.

Why does Discover Math use a investigative approach to learning?

Traditional math instruction, and a lot of the math instruction on-line, focuses on skills and memorization, rather than on understanding and applying. I believe that learning math skills and memorization of rules is not very important any more. (See discussion below.)

The investigative approach of Discover Math is designed to help learners understand the WHY behind the important math concepts we're learning and to see how those concepts can be used.

Why do you believe it is more important to focus more on concepts and ideas and less on skills and procedures?

In the "old days" (i.e., before the wide-spread availability of graphing calculators or graphing utilities), students had to be proficient at the basic skills or processes of math so they could then use those skills to examine the concepts and uses of math. For example, in calculus, it was important for students to be able first to find derivatives before they could use derivatives to analyze a function. The examples, worksheets, and homework problem sets were all specifically designed to make students learn the process, rather than understand and apply the concept.

This is exactly how math has changed dramatically in the past forty or more years. On-line graphing utilities, such as Desmos, can do the basic skills that students used to have to do by hand and that had to be drilled into students via examples and worksheets. Derivatives, integrals, and other math problems that used to require ugly algebra to solve can now be solved instantaneously, using tools that are literally right at their fingertips (and those tools are free!) Modern technology has all but obviated the need for students to focus on the process skills of mathematics.

Worse, the focus on drilling skills into students is counter-productive to real learning. Practicing skills is boring. Let's face it, no student comes into math class saying, "Oh please, can we have more worksheets?". To the extent students struggled with the algebra of problems, the algebra served as a barrier for them to continue learning real math. (This is one of the reasons, I believe, that calculus always served as a "weed-out" course in freshman year of college.) Discouraging students from learning about real math (i.e., concepts of math, not the skills of math) is a crying shame.

Don't misinterpret what I say here. A certain amount of mathematical skills are still necessary, but I believe the focus of teaching math should move away from skills. Learning how to do something just isn't has important as it was before. Learning how to apply or use a process is what is important in our modern era.

Traditional math instruction is pretty passive. The students sit in class and are expected to receive information from the teacher while trying not to zone out, thinking about anything other than math.

Similarly, a lot of on-line math instruction is basically passive. You watch a video (or read some on-line notes) of some one trying to teach you something.

With investigations and discovery, the students have to put in the effort right up front since there is little given to them at the start. The students have to work through the steps of the investigations and come up with the concept to be learned. The students also have to reflect back on what they did to make sure they did the investigation correctly.

Bluntly, the investigative approach of Discover Math is a lot more challenging than traditional instruction. But having something that is difficult sometimes makes it more interesting or engaging (in the literal sense of that word).

This form of learning is an off-shoot of what in the educational world is called “productive struggle”, the idea that students learn best when they have to work at it a bit.

Are there other ways where learning by investigation is better than traditional instruction?

Traditional math instruction is about mistake-avoidance. In traditional math instruction, mistakes are bad, and mistakes "cost" you points (making your grade suffer). Your teacher is "disappointed". You feel like a failure. And downward the spiral goes.

In the investigative approach, not only are mistakes to be expected, they are to be embraced. It's cliche to say that "Everybody makes mistakes," but it's nonetheless true. The real measure of a person is not whether one makes a mistake, it is whether one LEARNS from that mistake. Let's face it. Math is not heart surgery. If a heart surgeon makes a mistake, the result can be fatal. If a math student makes a mistake, you can look at what you did, take out your (electronic?) eraser, and start again.