The Parabola Less Traveled

A parabola curved in a yellow wood,
And sorry I could not follow both
For be one traveler, long I stood
And looked up one as far as I could
As it approached infinite growth;

Then took the other, just as fair,
And having, perhaps, a negative claim,
Because it rose left, and wanted wear;
Though for each x the y-axis there
Had been worn really about the same

This parody joyfully brought forth from my mind, for in fifth grade I was made to memorize this poem, and have never forgotten it. For reference, this is Robert Frost (please see the link here).

If I could teach anything in the whole, wide world, it would be Mathematical History.

When their first word is, "AAAAAAH," something's going wrong.

No, not the the history of mathematics, where we would follow the evolution of the subject, holding all else constant--which, as it were, is the basis for most multi-variable equations.

I know that for the in focus on, n≠(1/3)...

I'm talking about using mathematics as the lens through which look through time and ask questions about the societies that we see.

We would start our journey with a little question and discussion: is math--or, maths if you're not from the USA--invented or discovered? I'm very much in the later category, that mathematics is the underpinning of the universe and not a mere figment of the human imagination. However, I'm really curious what the education system will have taught the kids so far, because I'm pretty sure that they've never been asked this question. Instead, they've probably been taught the subject by people who didn't get it (this is one of the big reasons why I hear a lot of people want to teach the younger grades. They just don't get multiplication and beyond, but that's a discussion for another day).

We would then go to ancient Greece. Why not Babylon? (You ask, if you know anything about the history of math). Because, I would reply, I don't want to get into a year long rant about how base 12 is so much better computationally than base 10, and if we really wanted to make a race of super humans they'd have six fingers on each hand, and how cool that would be, and it would just go on and on and on. No, we'd start in Ancient Greece. We would read texts about ratios, inequalities, logic, and geometry. We'd ask how their cosmology affected the questions these people asked, and what problems they used their mathematics to solve. My students would be beaten over the head with the fact that Pythagoras is not a person, but the PYTHAGOREANS were a group of people who may or may not have worshiped mathematics as a better way to understand perfection.

We would then follow the texts as they migrated through the Roman--later Byzantine--empire, and ask how their mathematical  principles affected their art and architecture. Next, would be the Muslim caliphates--centering on Baghdad--with comparing how their cosmology affected not only how they developed mathematics, but how they interpreted the earlier Greek work. We'd also ask to what extent their culture--being far more cosmopolitan and less xenophobic (the word, 'barbarian' literally comes from the word, 'bar', which is how the Greeks said that everyone else talked. Political correctness in any form was not their forte. Nor were woman's rights, abolitionism, wearing pants, etc.)--allowed them to grow the subject higher than the Greeks.

We'd end our journey through the empire with a brief trip through the Cordoba Caliphate--specifically a man named Averroes and the mathematical marvel that is the Alhambra--and then re-enter what would later be called Italy just in time for the Renaissance (assuming a two-semester class). We'd then go through the enlightenment, and end with a brief discussion unit on how the pocket calculator--and computerized number crunching in general--has changed both our view of mathematics and where/ how it can be applied.

Throughout this history, we'd follow several themes. The invented/ discovered discussion would obviously play a part, but we'd also be looking at how each culture used math. What were the problems they were trying to solve? Why weren't the current tools good enough? How did they make new ones? The goal is to show that math (and other subject..hint hint, cough cough, History, hint hint, cough cough) was studied in order to be used. People didn't find these formulas to be taught in class, but to solve a problem, an aspect that is sadly overlooked in too many subjects.

The last thread is what I'll call spontaneous development, which is where distant groups--such as, in this case, the Chinese in Asia, or the Maya in America--developed the same concepts and techniques. I do this because I personally feel that the more view points I can add into a discussion, the richer the discussion is. It also gives some of these groups--once again, the Maya--more agency than they have traditionally been given. These were not just people waiting in an idyllic paradise for the Europeans to come and introduce, "modernity." These were people who had problems and were active in trying to solve them, sometimes discovering ideas that Europeans and Muslims wouldn't know about for at least a century more. It also opens the door for some interesting questions: if they were using theories long before their counterparts across the ocean, then who really discovered it first? Likewise, in the micro, if two people in one culture discover something at roughly the same time (such as calculus) then who should get credit? Just some thoughts.

This has been another exciting adventure in the Austentatious. If you liked it, tell your friends; if you hated it, tell your enemies; and if you don't care either way, then tell everyone.  

On How to Teach a Second Language

Every teacher likes to say that their subject is the most important. It's an argument that I've had time and time again with teachers teaching as diverse topics as mathematics to history, or from Spanish to (in one very odd conversation) drama. Call me a little odd, but I think that the methodology for teaching a subject is a better topic of conversation for teachers than the subject itself.

For example, why do we only seem to use word problems in math? Yes, there are long-essay prompts that are similar in many subject areas, but I can't think of any problems in any other class (except for maybe science) where I had to identify the necessary information in a block of text, and then use it in a multi-step process to solve a problem. In fact, the closest I've seen in Spanish is a series of, "Use the text to answer these following comprehension question," questions.

But let's say that you want to teach basic Spanish. What should you do? Well, there are many possible curricula. Too many, in fact, to really go in any kind of depth here. That's why, instead, I'm going to go through a slightly different process. I'm going to start at the end, and then apply it to what I know.

So, what's the end? What's the goal? What's the way for a 300-bowl?

To communicate, to talk, to understand, to converse, to produce.
In other words, to develop a communicative proficiency in both input and output in the target language. Or, in normal people talk, to be able to use the darn thing.

For any language, that means having a basic knowledge of vocabulary and grammar. Most people would include being able to read the letters, and--if you're feeling really hoity-toity--being able to write them too.

So, if the goal is to be able to use the language, how do we get our students to learn the vocabulary and grammar? Long vocab-lists and careful lessons on grammar?

Example A) Vocab List

1. Verduras
2. Frutas
3. Carne
4. Pescada
5. Pan
6. Manzana
7. Naranja
8. Platano
9. Lima
10. Lemón
11. Papa
12. Cebolla
13. Zanahoria
14. Berenjena
15. Pepino
16. Aguacate
17. Leche
18. Jugo
19. Chocolate Caliente
20. Presa
21. Comprar
22. Vender
23. Tienda
24. ect.

Example B) Sample Grammar Lesson

1. Direct Object Pronouns
1. Direct Object Pronouns are a set of words that refer to an object that the subject is interacting with. They are used to simplify speaking. For example, instead of repeatedly saying, "la manzana...la manzana...la manzana..." with pronouns we can just say, "la manzana...la...la..." Direct objects usually go before the verb. Depending on what the object is--me, te, lo/la, nos, os, los/las--the form changes. Here's a worksheet to practice working out the differences.
2. Now, onto when the direct object pronoun can go after the verb as a suffix--it's not required, but part of personal preference/ dialect--and when it must go after the verb.

At face value, both of these examples seem fairly effective. Students need a lot of vocabulary, so the teacher gives them lots of vocabulary and incentivizes learning it with a quiz. Likewise, the students will need to know the rules, so explicitly explaining them will help students pick them up.

The problem, though, is that people tend to learn best through repetition. Going back to math, that's why teacher's assign large numbers of problems. Of course, if there's no context for the problems, then students' can't really apply them, but that's just another layer.

So, where does that leave us?

If a vocab list is used, then make it short enough that all of the vocabulary can be practiced many times over the course of the week. If you'd rather not make a vocab list, it's always acceptable to elicit vocabulary from the students. Say that you're going to teach them about grocery stores; you can ask them what kind of food they like. If the students are interested in the words that they're learning, then they are more likely to not only use those words in class, but to use those words in their daily lives as well.

Likewise, I'm not sure that explaining the grammar really helps when students are first starting out with a language. My judgement might be clouded since much of my language teaching experience is with fairly low level students, but beginning students seem to respond better to learning either sentence frames that they can use or conjugated verbs, rather than the rules underlying why either those sentences work, or why the conjugation looks the way that it does. Comparing what my students with less traditional forms of language acquisition are able to produce with what I could produce at similar intervals, I don't see much value in an explicit explanation. Or, perhaps I should say, providing an explicit explanation at first. By letting students interact with the language, and try to make connections on their own, the teacher not only encourages the curiosity of the students, but respects their ability to make intelligent connections on their own. Once students have made a guess, then the teacher can suggest targeted corrections to help the students to understand the language.

To recap, it's important for a second (or third...or fourth...) language to be learned with the intent that, one day, it will be used. Students will need vocabulary and grammar, but need it in a way that is both accessible and will stick with them. Lastly--and this was only touched on at the start--it would behoove a teacher to look at methods that other subjects are using.

Some might indeed have wonderful ideas.

On Technology

I think that most teachers at some point in their career have heard the line, "technology boosts grades in the classroom." And, while that phrase can be true, I think that it requires a caveat at the end: "technology boosts grades in the classroom, provided that the teacher knows how to actually use it."

As with waiting, technology isn't easy. It requires a new way of approaching problems, and a new mindset. Unfortunately--both for most people older and younger than the cohort I grew up with--technology is not always intuitive. In other words, dear reader, please read that Tron lied, and technology will not give you easy to use superpowers.

A quick note on the, "younger" comment above: most students do not know how to use technology, they know how to be entertained by technology. Yes, compared to those with more gray hair than myself, their ability to take selfies, text, and watch a clip of a soccer goal--all while combing their hair, driving, and causing insane accidents in their wake--can seem magical. Mysterious. Almost...dare I say it...super human! But, ask them to do something with HTML--or, even worse, what WWW stands for--and many will give you the, "why on earth would you ever think that trivality is important, you incompetent nitwit," look.

First, some ideas of what to do with technology in a classroom:

1. Let there be an asymmetric perspective between what you are saying, and what's on the screen. If you only ever read off of the power-point, everything will be boring as heck, your students may feel insulted, and you'll only ever engage the students with the material in one way. By taking a more asymmetric approach, students will be able to engage with the material in more than one way, the show can be the main points while you can clarify/ go deeper, and I can guarantee that lessons will be more interesting for the students.
2. Do approach technology as something more than just an electronic whiteboard. The power of a computer is so much more than just a fancy whiteboard. It can look up information, change text on the fly, show videos, transport students halfway around the world, talk with almost anyone else in the world, play games, and almost any number of things imaginable. Explore what it can do for you, and you may be surprised by just how far it can take you.
3. Let technology be a shared experience. How can you get students to use their technology in a positive way in the classroom? How can you model good digital citizenship? How can you help your students to understand that their phone is more than a magical picture-taking box? The questions are endless, but so are the solutions.
4. Documentation. What should a teacher be constantly doing? Documenting and taking data. We're kind of like scientists, if bacteria could flip you the bird one second and call you the greatest teacher ever the next. Digital devices have a multitude of ways to document what's going on in class. You can take pictures of groups to remind you who belongs where, or pictures of projects to show parents what their child is doing in class. You can use voice recorder to capture conversations or discussions. You can take videos of presentations to re-watch later to either a) grade, or b) let students self assess. You can even put all of these together on a classroom blog or photo album to always remember the year together. No longer will you have to write everything down, let the computer do it for you.
5. Use technology as support for students who need additional scaffolds. Say, for example, that you have a student who can barely write. If you hook them up with voice capture technology--which, depending on how much it needs to be trained, can be fairly inexpensive--then they can write down their ideas without needing to pick up a pencil. Or, say that you have a student who is actively distracted by noise around them. Hook them up with headphones connected to a microphone you have on, and suddenly they can pay attention to every word that you say. Some students even need a brain break. Have a section of the room open for students to self monitor with some quick activities they can play while they reset. There are lots of opportunities.
6. Do teach a little bit of coding. It doesn't have to be anything fancy like HTML or Python--though I would like to say that HTML is here to stay, coding languages tend to rise and fall in popularity--but teaching kids coding helps them to understand how a computer works. A computer is, in essence, a perfect logic machine. It will only do what you tell it to do, know what you let it know, and believe what you want it to believe. By understanding this fundamental aspect of computers, students gain an idea about the basics of the apps they use as well, and gain a little practice thinking about logical sequences on the sly.

Second, some great resources:

1. Wikipedia: How many classrooms have banned Wikipedia? I don't know, but the reason's always the same: it's untrustworthy. Well, last time that I checked, that's a learning opportunity right there. Give students a Wikipedia article, and ask them to do a few things with it: 1) What parts are facts, and what parts are opinions? 2) How trustworthy would you say the sources are? 3) What biases can you see in the article? You can even use Wikipedia as an example of the stupidity of pedantics (the great editing war of, "Star Trek Into Darkness," vs, "Star Trek into Darkness" is a fine example), and--as a great way to practice reading for key words/ related words--have them play Wikipedia Golf (give the class an article to get to, and have them start with a random search. The person who gets to the target article in the fewest number of clicks wins). I've also found Wikipedia to be a great jumping off point for research, although not the best play to do all of the research.
2. Word Reference: A great translation tool if you are looking for a specific word in English or a romance language. It's even available on most smart phones.
3. Google Translate: *Gasp*, not the dreaded Google Translate! This site is notorious for it's mistranslated ("Yo reloj la tele" [I wrist-watch the television]), so I definitely don't recommend it for that. Rather, I like using it as a reverse-translator. Whenever I write something in a foreign language--particularly a sentence--I like to translate it back into English. With about 99.98% accuracy, if the resulting sentence doesn't quite make sense, then I've written it wrong.
4. Google Maps Street View: Are you talking with your students about the Taj Mahal? Open up Google Maps Street View and go there. Did you talk about the Alhambra in your Spanish class? Go and take a little tour. Wishing for warm, sandy beaches on a cold, cold day? Mele Kalikimaka is just a brain break away.
5. Google Docs: Excellent for both individual and group projects, you can see what students are doing while they are doing it. 
6. Pinterest: This one almost didn't make the cut because I don't use it, but I've heard great things from people who do. Basically, many of the best masters steal all of their best work, then make it their own. Why not start here?
7. YouTube: This is a great way to share videos. It's also a fantastic way to offer kids a summary of what they did that day to use either for review, or if they missed a day and don't want to fall too far behind. Using it to create a flipped classroom can also be nice...provided not everyone does it.

Lastly (because I prefer to look at things in the positive) some things not to do with technology in the classroom (note: These are largely a rehash of the above, but from a different point of view):

1. Don't read off your Power Point slide/ Google Doc/ what have you. Please, just...don't. I can read. By the time Power Point becomes a main tool in your repertoire, most of your students should be able to read. If you need help remembering, then use flash cards. If you read off of a slide, then you're a) insulting your students by implying that they can't read, b) wasting your student's time, because why should they listen to you when they can just read it?, and c) doing the students a monumental disservice. By reading off of the show, you're only giving your student one perspective of--and one explanation for--the material you are covering. By using a more asymmetric approach (and not writing in complete sentences) you'll be more likely to reach a wider variety of students who might need things explained in different ways.
2. Don't just use the projector like it's a fancy whiteboard, because then you'll only think of it in terms of what a whiteboard could do. Your computer can do so much more than just be written on, so please use it to your advantage.
3. Don't just think of technology as a teacher thing. Think about how to integrate it in a way that your students could use it.
4. Don't (read, DON'T) spend an entire semester/ year teaching kids how to type. Even those of use who know how to type have usually fallen into a slightly different mode, and most kids who don't learn how to type develop a system that works plenty fast for them. Teaching typing is utterly unnecessary, and focuses students to think about the use of technology, instead of the underlying mechanics that would make them really fun to use. Although, not everyone agrees with me (https://www.technologyreview.com/s/425018/out-of-touch-with-typing/).

To recap: technology is as amazingly creative as a student who has just realized they utterly forgot to do the essay due today and don't want to admit it. Use it well, and students can grow. Use it poorly and...well, it might be better to just stick with a whiteboard.