 # Other thoughts

Sometimes it's useful to provide distinctions at the beginning of a discussion. For example, when teaching logic, we can explain the distinction between deductive, inductive, and abductive reasoning, to give students a better idea of what they're about to study. That's because they've had experience with all the variants prior to the discussion. We have to separate the parts we're interested in studying from the parts we're not.

In other cases, it's best not to start with a distinction. For example, explaining the distinction between classical and intuitionistic logic should not be done at the start of a course on mathematical logic. That's because students are likely to only have experience with classical logic. There is no need to separate, because in their minds, classical logic is the only logic. The same goes for introducing non-Euclidean geometries. For most students, Euclidean geometry is all there is, so there's no need to explain non-Euclidean geometries until later in the course.

## Thoughts on the distinction of skills

A proper classification of mathematical skills requires careful distinction. At first, it may seem that solving linear equations graphically and solving quadratic equations graphically are two unique skills, but I wouldn't consider them so. If a student can identify the Cartesian coordinates of an intersection point, and they are familiar with graphs of functions, they can solve any function graphically. Solving linear systems graphically must be a separate skill, because students learning to solve linear equations graphically haven't seen linear systems yet, and won't be able to make sense of such a graph. In summary, we must consider the prerequisite skills for each skill, and we must consider the mathematical objects of which the student is familiar.

## Examples of perfect numbers, groups, etc.

Many math textbooks will teach a predicate by giving its definition followed by a few examples. Many computer science textbooks will teach an algorithm by giving an informal procedure, followed by stepping through the program for a few sample inputs. Why? Is there evidence to show that predicates having an infinite domain are better understood by looking at a finite number of inputs? Is there a proper number of examples in general? Could we converge on a good number of examples for a given topic by testing for comprehension?

## Designing a lesson

Before writing in a conversational style, the main ideas of a lesson should be listed. For example, what would the absolute value lesson include? Certainly, "the absolute value of x is the distance between x and 0 on the number line." Among the main ideas, we should include illustrations, animations, and interactives that would be useful. For example, a lesson on Riemann sums should definitely include a picture showing a graph and rectangles.

When choosing which videos to show for a lesson, we want to show all the main ideas. Preferably, we want to cover all the main ideas showing the fewest number of videos. However, this would be a dumb metric, as it's possible that one videos provides a better explanation for an idea than another. I am still thinking of how to formalize my choices of videos for lessons. Currently, my approach is to choose the videos I would watch if I were learning whatever concept for the first time.

## Helping students plan for far-off goals

Many of the things I've learned, I heard about a long time in advance. For example, I knew there was a topic called "real analysis" and that it was "a more rigorous version of calculus," long before I started learning real analysis. Is exposing far off ideas useful to students? Does it possibly help them to set goals, by making them ask "okay, so what would I have to learn to get to real analysis?"

## Surveying the mood of students

I want to foster life-long learners of math. To do this, I must provide enough challenge for math to be interested, but not so much as to make the student feel defeated. How can we survey the mood of students to enhance the curriculum?