*teaching is a system*.

By system I mean that teaching is a set of interdependent components that form an integrated whole. Instructor actions, course content, what students bring (intellectually) with them, and how students are assessed are some of the interdependent components. To simplify things we can focus on three main components of the teaching system:

- Teaching methodology (broadly defined)
- Course content and more specifically tasks students do in and out of class
- Assessment

Each of these components are complex, and have several subcomponents. I'll just touch on the surface of each of these in this post, as it is beyond the scope of this post to discuss all the important details. What I want to do here is to outline what I mean when I say that teaching is a system.

*Teaching Methodology*

In moving from teacher-centered to active, student-centered instruction, the instructor in an IBL (or hybrid IBL) course is to shift the focus from information transfer towards sense-making tasks. The instructor could employ student presentations, group-work, and think-pair-share as strategies in-class to engage students in the learning process. Teaching changes that involves students more deeply in the learning process implies that usual content has to be reformulated. Students need different tasks to work on compared to when they receive information from a lecture.

*Mathematical Tasks for Students (Content)*

In the google era the objective of education is no longer focused on knowledge acquisition. Knowing how to creatively use knowledge and effective ways of thinking rise in value. In light of this, merely sharing insights or showing how things work is not sufficient. Students then need different, more sophisticated tasks, which require them to think, problem solve, and communicate their ideas. These tasks then must be designed for the specific stages of development and the variance among students in each class. Selecting appropriate tasks relies on a knowledge of how students think, and tasks must adapt in sophistication as students develop over time.

*Assessment*

Measuring success in this new paradigm then needs to change. Merely asking students to execute algorithms or bubble in answers does not capture more sophisticated learning objectives, and does not provide proper incentives to become a better thinker. Thus, as one switches more and more towards full IBL methods, assessments must also change in parallel to measure appropriate learning outcomes. The simple way to put this is "Put your money where your mouth is." If we value problem solving, communication, creativity and all those wonderful qualities of an enlightened mind, then we need to align assessment accordingly. Specifically this means we need to add in presentations grades, adapt tests towards mastery (as opposed to showing knowledge acquisition), and course grades ought to move away from a weighted average of tests, homework, and final exam. Some of this is controversial, and I understand both sides of the argument, especially in the case of using a weighted average. My point here is really that instructors should ask themselves if whether a weighted average truly captures the learning they are after. There exists, in fact, an implicit acknowledgment that the weighted average is not always useful among many (or most) faculty. For example, it is common practice for instructors to weight the final exam more heavily if a student does better on the final.

Assessment doesn't always have to affect final grades -- informal assessment helps guide instruction. Interspersing activities in class to check for understanding can provide instructors with the information they need to address the specific misconceptions and learning challenges the students are dealing with. Not only are active, student-centered activities be better for learning, they provide a wealth of information for instructors to use to guide instruction.

*The Textbook is Not Enough*

Viewing teaching as a system can also help put things into perspective. For instance, textbooks are generally overvalued as a solution method for dealing with the issues of teaching and learning mathematics. In other words we may fool ourselves into thinking that we can solve our problems by just having "the right book." Indeed much of the reform movement has been focused on content, and for good reason. If we want students to know more than algorithms, then they need updated textbooks (or course materials). That's true. But let's say we get good materials into classes. All we've done is change one aspect of the teaching system, and consequently we need to consider if this materially affects the way in which students interact and engage in the mathematics. While choosing good course materials is important and necessary, it is by itself not sufficient. And this is clear from the teaching is a system viewpoint.

We can think about this issue another way. If it were the case that all we had to do is pick the right textbooks and then all our struggles with teaching and learning would be solved, then it would have been done by now. Throughout history there have been enough brilliant, eloquent writers and thinkers, who have provided us with a wealth of lucid, beautifully written work. Just because Shakespeare wrote

*Hamlet*(and that we read it in high school or college), this doesn't mean we have all learned, internalized, and implemented the valuable lessons presented in it.

*More Advanced Stuff*

There are other issues that come into play that haven't been touched on in this post. One has to consider the specific course (Calculus, Upper-Level Mathematics, Math for Elementary Teaching, Math for Liberal Arts), student attitudes and beliefs, getting students to buy-in, and so on. The teaching is a system framework provides us a broader approach to addressing some of these challenges.

*Teaching methodology, mathematical tasks, and assessment*can be adjusted and developed in coordination to shift the focus on student engagement and collaboration as well as addressing issues related to student attitudes and buy-in. More posts are coming soon that will present specific examples of how this can be done.