Author: mmccoo

TLDR: I try to provide what I wish I had.

I was a kid with a lot of curiosity. I engaged with the adults around me. I enjoyed learning and I extracted as much knowledge from those around me as I could.

But the people I knew didn’t really know that much math.

I have a memory from sixth grade. One day we had a substitute that I drilled with questions. Somewhere along the way, he showed me how to solve two variable linear equations using the substitution method. I went home and tried it with problems I made up.

But he was only there the one day and back then, we didn’t have YouTube.

Fast forward a bit to high school. My two main math teachers were very good. Ms London and Ms O’Bryan. Good classes, but they moved slowly. They went at the normal pace.1 One day, I show up to class and there was a quiz. I must have missed a couple days before because I didn’t know the material. So I asked the kid who sat next to me. Manny. “Hey, how do you do the stuff on this quiz”. He gave me a quick rundown 2 and I did fine on the quiz.

Now I’m pretty good at math, but I’d bet that every random group of 25 students has at least 5 students that could move at twice the pace or more.

Why not enable them to move at that speed?

Anyway, my senior year of high school, calculus wasn’t really offered. There were a couple of us that were labelled as calculus students, but it was in the same room and period as a normal math class. With the same teacher.

Ms O’Bryan already had enough on her plate teaching the main class so us calculus kids didn’t get much instruction or direction. I tried to learn from a textbook and I did make some progress. I managed derivatives and some basic integrals. I didn’t bother taking the AP exam. I would have bombed it.

Go into turbo mode.

So I land at MIT and my calculus knowledge lasted a lecture and a half and off we go.

I loved it. It was such a rush. The classes were hard but finally I was in a class that went at a pace that wasn’t boring. The class was 18.011 taught by Haines Greenspan and was probably the biggest influence to me majoring in math 3

Why did I have to wait til college to take a fun math class?

That’s what I want to provide to students.

But I’m not just about the gifted students

I’ve interacted with many “normal” students who were actually pretty good at math, but somewhere along the line their math confidence was shaken. Those students are often surrounded by adults who aren’t “good at math”.

They just need some encouragement and prodding. I’m pretty good at that too.

There are lots of people who say they’re not “good at math”. I argue that it started when they didn’t memorize the times tables.

Memorize. Not understand. Memorize

Think of the problem:

 34
x56
----

What does it take to solve this if you haven’t memorized the times tables.

ok. first I do 4×6. Right. What was 4×6 again? Let’s see 6+6 is 12. ok add another 6. um, 16? no 18. Another 6 makes 24? ok. Was that 4 times? I hope so.

ok, I put down the 4 and carry the two. Where does the two go again? (at this point the better part of a minute has gone by).

Ok, 3×6. Oh, I just did that one but I forgot the answer. 6+6=12. Add another 6. oh yeah right, 18. or was it 16?

Now where was I? put the 8 next to the 4. Oh, wait I need to carry the 2.

And so on.

The problem is that as you’re learning long multiplication, you keep getting sidetracked by basic multiplications.

• to keep losing where you were
• it takes longer to work the problem than the kid sitting next to you
• it makes long multiplication feel more difficult. There’s also this extra cognitive loading.

When you see 4×6, if you can just write down the 4, things go much smoother and your confidence doesn’t take a hit.

How about fractions?

\(\)$$\frac{1}{12} + \frac{1}{4}$$

When you look at the number 12, the numbers 3 and 4 hopefully sit there in the background. knowing that 12=3×4 means you can do

\(\)$$ \frac{1}{3*4} + \frac{1}{4} = \frac{1}{3*4} + \frac{3}{3*4} $$

instead of

\(\)$$ \frac{4}{4*12} + \frac{12}{4*12} $$

oh wait, what 4×12 again? 12+12=24, add another 12 to get 32? 38? No 36. another 12 gives 44? or is it 48?

Where was I? This fractions stuff is hard.

Factions are hard. They’re even harder if you don’t know the times tables.

You don’t have to be “good at math” to be learn the times tables. You only need a functional memory. Learning them is a real drag. I hated it when I learned them. I really didn’t want to study them. I hated memorizing anything. I hated spelling too. but my parent forced me anyway.

There are very few things schools make you just memorize. times tables and the 50 states. That’s about it. You just have to learn them.

How about some algebra

\(\)$$ 3(7x+3) $$

Ok. I’ve got this one. First I take the 3 and multiply it by the 7. right. what’s 3×7? 7+7=16 or is it 14. it’s 14. Add another 7 to get 21.

Now, where was I? ok 21x. What do I do next? I forgot. oh yeah I add the 3. The answer is 21x + 3.

Wrong, forgot to also multiple the 3 by 3.

but if you just know the times tables, you quickly get to the point where you write the answer without even thinking about this.

What about

\(\)$$ \frac{3x+12}{x+4} $$

If you know that 12 has a 3 and a 4 in it, it’s much easier to get to

\(\)$$ \frac{3(x+4}{x+4} $$

enabling you to cancel the x+4.

All if this is easy or hard based on whether the times tables have been memorized.

My number one advice for any parents: learn the times tables and then make your kids memorize them too.

One could also ask Why math remediation? No one questions that. We should meet everyone where they are. Also, acceleration opens some possibilities.

Here’s the normal progression of math for students in PPS schools:

• Algebra 1/2. one and two variable linear equations. function notation, interpretation. Basic function scaling and x/y shifting. Introduction to exponential growth/decay. Introduction to quadratics.
• Geometry/Statistics. Basic trigonometry, triangles, averages, mean.
• Algebra 3/4. review algebra 1/2. arithmetic, geometric, and recursive series. more advanced function concepts
• Precalculus. review algebra 1/2/3/4 and trig. more complex function scaling. logarithms. extend quadratics to polynomials and rational polynomials
• AP Calculus

That’s five years to calculus but high school is only four.

There are two PPS sanctioned paths to AP calculus

• some middle schools offer algebra 1/2 in 8th grade.
• in high school algebra 1/2 and geometry can be taken concurrently.

Cool. what’s the problem?

For motivated kids I see a couple problems.

• Until you get to precalculus, the classes are slow and boring. We’re talking about motivated kids. Doubling up means they spend twice as much time in slow and boring classrooms.
• Precalculus doesn’t really prepare you for AP Calculus.
• No possibility to take advanced calculus.

How do I define motivated

When I talk about motivated kids, I don’t mean the really exceptional, prodigy kids. For me, a motivated kid is someone who:

• shows up to class ready to learn.
• They’re able to basically sit still and actually pay attention.
• Doesn’t have large gaps in prerequisite knowledge.
• Thinks math is at least a little interesting.
• I’d guess this is at least 10% of students.1

Ready to learn as in they don’t mind being at school. Their goals don’t include sidetracking the teacher. They are still a little curious about the world around them.

Some kids, especially boys, need to bounce around and burn energy. They have difficulty focusing. There are strategies to mitigate this.

The biggest issue for many is they haven’t learned the basics. In particular, times tables and fractions. Gaps like these make it difficult for otherwise bright kids to succeed in more advanced math subjects.

Classes are slow

Algebra 1/2 and 3/4 cover mostly the same concepts, just a bit more depth. These concepts can be challenging to many or even most. Motivated kids could learn both subjects in one year. Really motivated kids could also learn precalculus in the same year.

For kids that can handle two to three times the pace, normal offerings are boring and they’ll wonder whether they like math.

I’ve heard a reason not to accelerate that I find peculiar. The reason is that kids that are accelerated early will often not take more advanced classes later in their schooling. Here’s my take on that: the kids are tired of being bored. Even being bumped up didn’t fix it. Instead, course offerings should be more compelling.

If a student excels in math, they should be rewarded with more challenging classes.

precalculus doesn’t really prepare you for AP Calculus

At some PPS schools, students can earn PCC precalculus credit… for one semester. PCC has two before their calculus. In addition to this, AP Calculus, even the AB version, is pretty challenging. So we’re expecting students to switch from a high school paced bunch of classes and then the next school year they’ll go into turbo mode to prepare for the AP exam? Hardly.

Taking an AP class and doing well (3+) on the exam are two different things.

If a student is accelerated they’ll get to more challenging classes earlier. Knowing that many students 2 will run out of coursework high schools could offer more rigorous options.

The benefits of acceleration take a while to manifest

In PPS, fifth or sixth graders can be bumped one year ahead. So a fifth grader can be in a sixth grade math class or a sixth grader can be in seventh grade math.

PPS requirements for this can be found here.

For a kid who manages to accelerate, those sixth and seventh grade math classes are just as boring as what they experienced before. No real benefit there, but…

They will be able to take algebra 1/2 as seventh graders and geometry in the eighth grade. PPS offers, begrudgingly, a self paced, Khan style course for geometry. Algebra 3/4 as freshmen, Precalculus as sophomore and AP Calculus as a junior.

They’ll have something to report on their college applications

Also, they can take advanced calculus as PCC, PSU, or UoP.

But really, acceleration should be more aggressive so motivated students can experience the joy and the rush of being in real classes much sooner.

Math is awesome, but our system tries to convince students otherwise.