### Improving the math chapter

The goal for the NO BULLSHIT guide to MATH & PHYSICS was to make a concise textbook that teaches university-level calculus and mechanics in a nice “combined package.” The math fundamentals chapter grew out of the need to introduce the prerequisite material that many students often lack. I didn’t want to be like “y’all should remember this math from high school,” because if you don’t remember the material such comments would not be very helpful. A review of high school math would be more helpful.

Over time, I kept adding and improving the introductory math material in Chapter 1 until it reached the point that it’s a pretty solid little intro to high school math. I was very proud of the fast paced flow of explanations which manages to cover a lot of material (70% of high school math topics) in less than one hundred pages. Many readers also praised this chapter, saying how useful they found it as a review of high school math topics.

Recently I’ve been hearing from several readers who say the intro chapter sucks, and the book sucks, and by extension I suck. If it was one or two reviewers I could have dismissed this feedback, but now I realize there is a clear and consistent message in the readers’ feedback: Chapter 1 sucks as a first contact with math. My effort to “cover” all the high school topics in a fast-paced narrative like in the free mechanics and linear algebra tutorials is probably the worst thing to do for absolute beginners. I can totally understand why a reader who is not familiar at all with sets, algebra, and functions will have a rough time in the opening pages of the book. In the words of a reader, the book “goes from 0 to 60 in the blink of an eye,” which might be a good thing for a sports car, but not for a math book. It doesn’t help that I say “anyone can learn math from this book, regardless of their mathematical background” in the marketing copy. I need to do something to fix Chapter 1, and soon.

So what am I going to do about it, then? Write, of course—what else can a writer do? I’m going to prioritize the basicmath project and write the best sequence of introductory math lessons that ever existed! I’ll then use these explanations to beef up  Chapter 1 to make it a solid foundation. I think adding 20–40 more pages will be enough, so the book won’t get that much thicker. It’s not just about adding though, I think Chapter 1 could use better organization, flow, and clarity of explanations.

Interestingly, the basicmath project overlaps well with my planned social media campaigns that will push the message “learn math; math is useful,” as well as the math lessons by email. February is gonna be very mathematical!

### Learning can be fun

I just read this excellent article Pragmatic Learning: It’s not “fun” on Roger Schank’s blog. It’s a very good post that calls bullshit on the “gamification” cargo cult which is widespread in the edtech and corporate training world. Just adding points, badges, and levels to a corporate training program that teaches you something boring is not going to suddenly make it fun. The author’s main observation is that forced learning is not fun and we need not pretend it is. Consider an employer who wants their employees to know X because it is required by law, or a bunch of students forced to learn Y or else they’ll fail. These “forced” trainings are not fun, and gamifying them is akin to putting lipstick on a pig.

Instead of gamification, the author suggests learner’s experience should focus more on things like:

• Getting away from the one-size fits all approach:

Courses need not be administered to multitudes. One can have a course that is for one person only and can be used when needed. […]

• The use of simulators
• Enable students to collaborate with peers who are learning the same thing
• Have human teachers (tutors) available to help
• Enable “learn by doing” experiences:

[…] real autonomous, motivated, learning happens when you are in the middle of doing something, and questions arise in your mind about it.

• Provide training in a “just in time”(JIT) manner, e.g. provide training on X right before the student will need to do X.

I highly recommend you read the article because the above summary hardly does it justice. I agree with 90% of the observations in this article, but I have some comments and observations of my own to add below the fold.

### Abstract

Calculus and mechanics are often taught as separate subjects. It shouldn’t be like that. If you learn calculus without mechanics, it will be boring. If you learn physics without calculus, you won’t truly understand.

I think I may have found a way to solve this chicken and egg problem. It goes a little something like this:

1. Chapter 1. You need [solving_equations,algebra,quadratic_equation] to do physics. That is all the prerequisites for first year Physics.
2. Chapter 2. Physics laws are expressed as equations. If you know how to solve equations, then you know how to solve physics equations. In particular we will study the kinematics equations x(t), v(t), a(t) which describe  the motion of an objects.
• The kinematics concepts are: t, x(t), v(t), a(t), x_i, v_i. The equations of UAM are: a(t)=a, v(t)=v_i+at, x(t)=x_i+v_it+0.5at^2. Definition: free fall, an object on which only the force of gravity acts. Such objects experience a constant downwards acceleration of magnitude 9.81[m/s^2].  Examples: ball falling, car acceleration, ball being thrown downward. Equations are cool, but where do they come from? In order to find out we must take a short excursion into calculus-land.
• Calculus is the study of functions. We use calculus in order to do describe how quantities change over time (derivatives $\frac{d}{dt}$) or to find the total amount of quantities that vary over time (integration $\int \cdot\;dt$). Integral as an area. Examples: f(t)=3, F(t)=3t. g(t)=t, G(t)=0.5t^2. But why integrals?
• Integrals are the inverse operation of the derivative. In analogy with the techniques in [solving_equations] where we applied the inverse functions to solve equation and find the unknown number, we can apply the integral operation in order to undo a derivative operations in equations.
• Since we have been studying kinematics in this chapter, we now get to see where the kinematics equations of motion come from. We start from F=ma. Recall that our goal is to find x(t). Let us therefore rewrite F=ma(t) so that we can see the x in there F=mx”(t). F is not equal to x. F is equal to the second derivative of x. But no problem. We just learned that integrals are the inverse operation of derivatives, so if we want to solve for x(t) in F=mx”(t) we can do it! First we divide both sides by m, in order to isolate the x expression on the right F/m = x”(t). Then apply the integration operation twice in order to undo the two derivative operations.
• In particular, let us consider the case when F=const and m=const which implies that  then a(t)=const=a. The equation we have to solve is F/m = a=x”(t). Applying the integration operation to both sides of this equation we get at+C=x'(t). By definition x'(t)=v(t) so the constant C can be identified as the initial value v(0)=v_i. Applying the integration operation to both sides a second time gives us 0.5at^2 + v_it + x_i = x(t). This is how the UAM equations are derived: F=ma and two integration steps.
• Main idea of this book: understand the math + physics is easier than just learning physics by memorizing the equations. With memorization, you would need to remember three equations of motion as separate entities. If you understand derivatives and integrals then you can remember just one equation a(t)=a, which is not much to remember since it is in the name UAM.
• We have now seen kinematics in one dimension. But the real world is three dimensional so we need to learn about the math for dealing with objects in 3D.
3. Chapter 3:Vectors.
4. Chapter 4: Now that we know about vectors we can discuss more physics.
• Projectile motion. The position of the object is now a vector r(t)=[x(t),y(t)]. There are two separate sets of kinematics equations. x(t) is UVM (since no forces in the hz direction) while y(t) is UAM (a_y=-9.81 due to the force of gravity).
• Introduce dynamics F=ma. Forces cause acceleration. Forces. Force diagrams.
• Momentum.
• Energy.
• Uniform circular motion.
• Angular motion.
• SHM.

The structure in Chapter 2 is the only new thing. After that, Chapter 4 is pretty much a standard course through the mechanics curriculum. So how is Chapter 2 so special, as to be worth blogging about at 2:44 in the morning?

I will tell you in point form, because it is kind of late indeed:

• It connects nicely with the Precalculus chapter. You just learned how to solve equations for 50 pages, and now I am telling you that you can do physics with this equation solving skill. Yey! Math is useful.
• Then we introduce a bit of basic kinematics concepts x, v, a and the equations of motion. But then we say where did these equations come from (this is kind of a weak point?). To tell you, we must learn Calculus.
• Bam — drette là — we do a mini course on calculus in 5 pages. Integrals with pictures and FTC. Sure it is complicated but the analogy to f and f-inverse should make it go through.
• Then show derivation of x(t) via int( int(F/m) ). I can use the exact integral formula since students just saw those formulas as pictures 4 pages ago so they can’t say “i don’t know integrals”.
• Having this early exposure to integrals also helps with the work and potential energy section later on in Chapter 4.
• Basically the 5-page mini introduction to integral calculus is sufficient to do calculus-based mechanics course. The sin/cos derivative info and the chain rule required for deriving the SHM is presented in a just-in-time manner (i.e. in the last chapter).

But who am-I to say what is and isn’t a good way to teach. Only the students can tell me.