An essential proof

I've already mentioned the need for proof at the high school level. Proof is the essence of mathematics and the power of deductive thinking, such as in Euclid's proof that the primes are infinite, can help better students appreciate mathematics.

Most students, however, need an even simpler theorem to learn:

Theorem: Let p and q be positive integers. If N=pq then either $latex p \leq \sqrt{N}$ or  $latex r \leq \sqrt{N}$

I've written the proof (by contradiction) of this trivial result on the Handouts page. This is, in my opinion, an essential proof for high school mathematics because not only is it easy to understand, it's useful as well. Every high school student has to learn how to factor an integer into a product of primes and many of them try consecutive integers 2, 3, 4, 5, 6,.... But the theorem above tells us how to quickly determine whether a number is prime. So not only can the proof be motivated, but students get to enjoy the benefits of the result.

So, for example, if we want to figure out whether 149 is prime then we only have to check whether it is divisible by numbers less than $latex \sqrt{149}$. But there's no need to check composite numbers either because, for example, if 149 is divisible by 6 then it is divisible by 2 (and 3). Therefore, to test whether 149 is prime we only have to check divisibility by the prime numbers less than  $latex \sqrt{149}$: 2, 3, 5, 7, or 11. This is a basic result that most students (and even some teachers) aren't aware of.

A useful mistake

When I was in college I learned that calculators made mistakes in graphing. Now, it's decades later and it's still possible to find these mistakes from time to time. The most useful mistake is with the graph of $latex y=x^{1/3}$, the inverse of $latex y=x^{3}$, because it fits nicely into the high school curriculum. The calculator in the screenshot above is freely available on the internet and shows the mistake produced by graphing $latex y=x^{1/3}$. You can see this calculator for yourself at the Cool Math website. When I was teaching overseas, I designed a worksheet that relied on this graphing calculator to illustrate some of the flaws of technology using this calculator. I wanted them to see that they couldn't just blindly trust the results that they got from any calculator. That worksheet was lost in the Great Move back to the US, but the basic flow was something like this.

The students were studying inverses in precalculus and the worksheet had them:

  1. Graph $latex y=x^{3}$ using the graphing calculator, then sketch the result onto the worksheet. What is the domain? What is the range?
  2. Prove that $latex y=x^{3}$ is a 1-1 function.
  3. Since 1-1 functions have inverses, find the inverse of it using the procedure we learned in class (Swap y with x and solve for y). What is the domain? What is the range.
  4. Plot the graph of  $latex y=x^{1/3}$ (the inverse they found in 3) along with  $latex y=x$ and  $latex y=x^{3}$.
  5. Note that something is definitely wrong: the picture is not the complete reflection of the graph around the line $latex y=x$ and the domain and range of the inverse didn't switch places like they were supposed to. (Some students plotted y=x^1/3 creating a line, yet another lesson learned on the need to be precise with technology).
  6. Have them fill in a table for the function $latex y=x^3$ for values $latex x=-2,-1,0,1,2$ and using the fact that $latex (x,y)$ on a function means $latex (y,x)$ is on the inverse they can fill in a table for the inverse.
  7. Plot the results (by hand) onto a graph. They'll find out what the correct answer should have been.
  8. Now that we know the points that we should have gotten, have them calculate $latex (-1)^{1/3}$ in the calculator. They'll get the answer NaN (which stands for "Not a Number"); yet they can easily see it must be -1. The calculator is wrong again!

The worksheet became a way to emphasize the teaching point, "Calculators are tools which require mathematical skill to use properly." and impress upon them the importance of checking the results. It's a message I tried to deliver again and again, and I think you'll find it's an important lesson whether you decide to make it a worksheet, slideshow, or demonstration.

Education for the masses

I used to enjoy teaching, back when I was in private school. I thought nothing of spending my weekends and summers working on ways to make math a little more interesting and unusual for my students. Now that I'm working in a public school the experience is very, very different. It's "education for the masses" (my term, not theirs) if you will and it's not very rewarding. That's due, in part, to the fact that in private school I could contribute to picking the book and had a lot of autonomy in deciding how to teach the curriculum. Sure, my semester exams had to be approved content and difficulty. In public school I'm micro-managed. I have no control over the seating or even how to teach and they're even out to control the basic flow of every class with:

  1. flipped classroom
  2. Videos to introduce students to concepts before they are taught in class.
  3. Online quizzes through Edmodo (true/false, multiple choice, short answer)
  4. Online testing through sites like Quest and Quia.
  5. Lots more standardized testing.
  6. Targeted teaching of material to various groups within the classroom.

The basic idea is that students learn material before it's discussed in class through videos with a short Edmodo quiz to make sure they've done the assignment and perhaps a problem or two to do before class. From their performance you put them into groups on how they've done and then you visit each group so you can give explanations appropriate to students knowing what they know. When you've communicated what you need to then it's off to the next group to help them. Online testing involves the teacher picking problems from a problem database and assigning a window of time (eg 24 hours) over which the students can take the quizzes/test. That's done outside of class, leaving even more time to spend on learning material. In fact, the computer will score the tests, saving the teacher from having to grade all those tests.

I call it "education for the masses" because of the online testing and the forced structure of how you're required to teach. In private school I could lecture one day and do an activity the next day---whatever I decided was the appropriate method. Now that decision is being taken out of my hands, made by some nameless, faceless (brainless?) bureacratic with no meaningful math background mandating how teachers of all subjects must act. Until the next 5 year plan. Got it, comrade? It's a production like process to "educate the masses" and, in my opinion, the most stupid and unprofessional aspect of the system involves students taking online quizzes/tests which are graded by the system as right or wrong. With online tests

  1. You're not teaching students the process of solving a problem logically, step by step. The only value is given to the correct answer.
  2. Without the work, a student won't know where they made the mistake so it is harder to correct.
  3. Without work, a student can get the correct answer for the wrong reason and the teacher isn't able to correct it.
  4. You can't guarantee that the person taking the test is your student and/or that they aren't receiving help.
  5. As a teacher, you have less understanding for how well the student knows the material because you aren't grading their work. A computer is.

Showing work, which should be the of the utmost importance, especially in math, is meaningless in "education for the masses". To illustrate point 3, imagine the student has the problem -2^3 and gets the answer -8; they must know what they're doing because they got the problem correct, right? Not necessarily. If they were forced to show their work and wrote $latex -2^3=(-2)(-2)(-2)=(4)(-2)=-8$ then as a teacher I can correct the mistake: the base is 2, not -2. For more complicated problems it's easy for errors of understanding to remain hidden if you only examine the answer and not the work. As a private school teacher I forced my students to show me their steps to get full credit. This allowed me to catch mistakes in their mathematical thinking process (before they practiced the wrong way over and over). It also helped to cut down on cheating because getting the correct answer only got them 25-33% of the points for the problem. In public school, the larger number of students makes such individual attention impossible. As a teacher, I find that incredibly frustrating.

When I mentioned that we can't guarantee the student has taken the test on their own I was told, "they're only cheating themselves". Don't be surprised then, when they do as recently happened with the SAT in Long Island. It's just a matter of time before some student is on TV giving an interview how they graduated from school despite being mathematically illiterate because they found a way to cheat. Shouldn't we as teachers be striving to give more time to helping our students and making cheating more difficult? As a teacher it's hard to believe we should be building a system that encourages cheating.

If the online testing is the stupidest aspect of the model, the videos are the most obnoxious. The message is clear: find videos that the students can watch. And, by the way, avoid Khan Academy because that's just a guy lecturing with a tablet. Oh Sal, how you've fallen. Not that long ago you were creating the videos that I had to use but (perhaps because of your comments about minimal preparation [here and here]) now you're an example of the wrong way to do things. But back to my point: the message isn't "here are some great videos to teach X", it's get videos (and if you can't find them, we're told, then you can "make your own"). There seems to be no regard for educational value...and what ever happened to giving teachers the resources they need to teach?

What's most offensive about the "education for the masses" crowd is the contempt for other forms of teaching. All methods of teaching have positive and negative aspects. I already pointed out some flaws in their model but some obvious questions should be asked.

  1. What's wrong with other methods of instruction?
  2. Why is this method of instruction preferred over other methods? Top schools don't enforce this educational model.
  3. Colleges and universities have a more lecture style classes because lecture style is good for presenting large amounts of information. These college teachers have a mastery of their field that the public school teacher doesn't. Why are they wrong to teach lecture style?
  4. Why are teachers micromanaged by people who lack the knowledge and experience  that the teacher has?
  5. At a time when budgets are squeezed a lot of money is being spent on technology and proprietary programs (rather than open source).  Why is this a top priority when countries such as Shanghai are clearly able to better with less?

"Education for the masses" is one of many valid teaching models but the heavy handed way it is foisted upon teachers is ignorant and misguided. You can see big business getting involved through tablet computers and online testing (making things expensive) and when you consider a teacher adopting the model avoids having to grade papers, it seems "education for the masses" is here to stay.

For all the stupidity in the public school system I want to point out something that the public system has had incredible success with: following the example of private schools. If you look at top public high schools you'll find they have entrance exams that students need to pass to get in. Once there, they can still be removed for misbehavior or poor academic performance. Being allowed to set a level of performance and remove students solves so many problems because it sets a quality level. Poor students can't drag down the rest of the class. It keeps helicopter parents in line and lets the school take control in deciding which students should go into an honors class. Students and parents will settle on a school where they see a value (and work to stay there). After that, these exceptional schools hire teachers with masters degrees and doctorates and keep class sizes small so that the students get more individual attention.

If schools really want to improve, the path has been shown. More exam requirements to attend better schools and make it easier for experts in the field to teach. Unfortunately, the system has purposefully created rules and regulations to keep passionate experts of a subject out of the classroom (otherwise why would someone spend the time and money to go to a teaching college?). They've defined the word "qualified" so that it only applies to someone with a teaching certificate. As a result, they can stand there and say they are having trouble getting qualified math and science teachers--thye've excluded the really qualified people (in the common sense form of the word). This is shameful and needs to change; but don't count on them to fix it voluntarily.

Perhaps, like so many other education fads before it, "education for the masses" will fade away, maybe even for the reasons I explained. But I fear it won't; saving teachers from having to grade tests and quizzes means it will have a lot of support for a long time to come. What is clear is that 5 years from now, the typical public school education and there will be people extolling the virtues of a new educational system.

Golden Rules

Every student will be more successful at math if they follow the Golden Rules of Mathematics that I've put together:

Rule 1: Keep up with the material
You can’t expect to learn things just before a test and you risk some unexpected problem that prevents you from studying as you planned.
Rule 2: Pay attention in class
If you understand what we do in class then you’ll be able to do the homework more easily.
Rule 3: Do all the homework/classwork
At a minimum you should do every problem assigned for classwork and homework.
Rule 4: Show all your steps
to get full credit on tests, quizzes, and assignments.
Rule 5: Ask for help
when you can’t get the right answer or don’t understand what’s been said in class.

You can find a tex file and PDF version on the Handouts page.

Integrals and Series

When I was taking calculus in high school, I never really felt comfortable when I read something like $latex \int_a^bf(x)dx=\lim_{n\to\infty}\sum_{i=1}^nf(c_i)(\Delta x)$ in my book. Sure, I understood the basic idea about upper and lower Riemann sums but something more concrete would have been helpful. Nowadays software can quickly determine upper and lower Riemann sums easily (here's a Sage interact manipulative) but I'm talking about something even more: an example with an honest-to-goodness formula as well as limits to calculate.

That's not easy to do (because we need the series to have a nice closed form expression) but it is possible: the curve $latex y=x^2$ uses the series formula for $latex \sum_{i=1}^{n}i^2$ (which students should know and will result in a closed form expression) and requires the evaluation of limits from using the Squeeze Theorem . I've used it in previous accelerated calculus courses and have been happy with the results.

You can find the details on the Handouts page.