I wrote, in what is now (thanks to Jason Buell) the most famous of my small collection of blog posts, that:

Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly.

*Recently, my attention was brought to three excellent posts on story telling by Grace Chen, Dan Meyer and Dan Meyer (man, what are the odds those last two guys would have the same name!), all of which you should read before this post. In fact, if in the course of reading those three you get so tired or inspired as to not want to read my follow-up, then consider yourself suitably enlightened and come back later; what I’ll write certainly isn’t earth-shaking or particularly useful (nor will I be brief). However, I will tell stories and I will use bold fonted section names that arbitrarily oscillate between my voice and the voice of the reader held hostage in my head.*

# Part 1

At some point, my wife and I decided we would homeschool our children. Thankfully, we’ve had a good deal of time to prepare; deciding between the plethora of homeschooling structures and then choosing the specific curricula is a daunting task. We finally settled on the classical method popularized recently in The Well Trained Mind. Part of this program is to break up the child’s education into three phases: grammar, logic and rhetoric, roughly corresponding to 1st-4th, 5th-8th and 9th-12th grades. Within each phase, you study world history from the ancients to the present. So as to be practical, you probably don’t start at the beginning of history, but at around 5000 B.C.

While my oldest son is only four and so a little young to start the world history curriculum, he is old enough to begin his video game curriculum (and if you doubt the importance of this aspect of learning, please do watch Jane McGonigal’s informative TED talk). Sure, I could start him on modern gaming systems (he does enjoy Wii Sports quite a bit), perhaps spend $50 on Super Mario Galaxy 2, but it really wouldn’t give him a full picture of the gamer universe, now would it?

The answer: start at the beginning. Well, not really. Much like ancient history, we don’t want to start too early. Although I did enjoy playing games on the Commodore 64, the TRS-80 Color Computer and the Atari 2600, to me gaming as a civilized hobby begins with the Nintendo Entertainment System. Naturally, we started with the original Super Mario Bros. The exploits of plumber brothers Mario and Luigi quickly enamored my son, Alex. Soon, we were watching episodes of the Super Mario Bros. Super Show! and now both kids want a Mario themed birthday party. Good work, Dad. During one episode, there is a trailer for the Legend of Zelda animated series. My kids were hooked and Alex wanted to play the game.

**Your 4-year old is playing Legend of Zelda?**

No, not really. He sits on my lap while I play. So here we are, a few weeks later, just one level away from entering Death Mountain to fight the evil wizard Ganon for the Triforce of Power and control of Hyrule. I remember playing it as a kid and it took months for me to unlock all of the hidden loot, destroy all of the bosses, get the magic sword, figure out the NWSW maze trick, etc. It is a wonderful game that kept my friends and I occupied for a *long* time. Ah, but now I am older and the age of the interweb is upon us! I don’t have the time to search through Hyrule for everything again. No, I simply googl’d “Legend of Zelda walk-through” and found this delightful site which not only explains an efficient way to work through the game, but also has videos of the author doing it.

**Isn’t that cheating? **

Well, of course it is cheating.

**So, you’re teaching your son to cheat?**

Wha-buh-guh!!! Yes, that is the point. If studying math has taught me anything, it is why struggle to learn things when others have struggled before? Stand on the shoulders of giants and pick the (now) low-hanging fruit. It took people thousands of years to come up with modern mathematics; why reinvent the wheel?

**…seriously?**

**Part 2**

Is it better to know a few things really well or a lot of things reasonably well? I suppose it depends on what you want to do with your time. However, either is preferable to not knowing anything at all. Here is an experiment:

- Repeat the number 8 one hundred times in a row to somebody. At the end (assuming they didn’t leave or punch you in the nose) ask them what number you were saying. Chances of success: extremely high.
- Now ask (for your sake) a different person after repeating 38502 twenty times in a row (still 100 numbers spoken), what five numbers you were saying. Chances of success: high.
- Find a third person and tell them any random string of one hundred numbers and ask them to repeat it. Chances of success: essentially zero.
- Ask them about the first number of the sequence. Chances of success: low.
- Ask them about the first five numbers of the sequence. Chances of success: essentially zero.

**So what?**

So what? This is what our math curriculum feels like sometimes: a seemingly random ordered sequence of topics with often arbitrary repetitions that leave an unmotivated student with, essentially, zero chance of success. Our smarter students can see through the games and learn to hate math; our weaker students just learn to hate themselves.

**Standard Based Grading to the rescue!**

No! SBG isn’t a panacea. It is like watching sports in high-def: yes, you get to see Tom Brady’s nose hair, but a 42-10 rout still isn’t much fun to watch. The problem is with the game, itself. At several points, standards have come down from various organizations as if they were commandments from Mount Sinai. At best, I gather that someone took a reasonable modern calculus text and backtracked to decide what and when students should learn. Of course, one can’t simply do that. This wouldn’t take into account current practice, teaching training, etc., so you would have to create standards that compromised a bit to deal with reality. Does this make a good story? Hardly. The student is told: follow these instructions to the promised land. The student hears: Wa wa wa wa. The student learns: Nada Nada Nada.

**A Tale of Two Stories**

Much like the Star Wars Trilogy (x2) where there is an overarching storyline as well as individual episode with subplots, teachers also need to be aware of the overall story of mathematical education along with the day-to-day practice of teaching concepts, methods and algorithms. While the posts of Grace and Dan are geared at addressing the second part, the first part is equally important.

Currently, we teach from a walk-through. There is no sense of exploration and no sense of importance. Everything is deemed important except for those things that are difficult to test. While I think of mathematics as a subject that should bring order, we teach nothing but chaos. Ask a high-school student what precalculus is about. Hah! You might as well ask them what the dictionary is about. It seems we’re afraid to make the difficult decisions, afraid to cut down the quantity for fear that they’ll need it somehow, somewhere, sometime.

**But here is the funny thing…**

At some point, a student will ask their teacher: what is this all good for? One variation of the answer is to say: while this specific topic may not ever be relevant to you, the skills you learn from the process are. Okay, so some of us think that mathematics helps develop critical thinking skills and that the material is simply a catalyst. On the one hand, we’re almost willing to concede that some specific things they learn probably aren’t relevant to their lives, but, on the other hand, are terrified that they might worry about how awful it would be if we didn’t teach it. But, if mathematics really does help develop critical thinking skills, then it really shouldn’t matter if we skip some material: they can learn it later if they need it.

# Part 3

I play a lot of board games and one of my favorites is Race for the Galaxy. Really, I love hand-management type games. There is nothing more frustrating than having to choose between two awesome cards, knowing you’ll have discard the one you don’t choose. But, the decisions must be made and the game will go on. Educators in charge of curriculum design might first play these games and get used to making these types of choices. Imagine if you could only teach a student three things. What would they be? Obviously, your subject and level dictates this choice a bit. You wouldn’t teach multiplication to students who already know calculus, for instance. As someone who trains mathematics majors (many of whom will become mathematics teachers), I choose the following three facts (along with the appropriate discussion of what those facts mean):

- 1 is not a prime number
- 0 is an even number
- Derivatives describe rates of change

If you let me teach math majors about just the first two, I think I could put together a pretty solid class; the third is gravy.

**Where is the story?**

The story begins with a choice: what do my students *need* to learn. Right now, there are too many characters and not enough character development; too many storylines and too few stories; too much mathematics and too little opportunity to appreciate mathematics. Throw out what you would like them to learn or what you think future teachers would want them to know. Focus on the essentials. Now teach it, tell it, break it down, build it up, grade it, shake it, bake it, use it, love it, leave it, come back to it and, finally, they will know it. Let the students explore and let the students struggle. Don’t help them, help them, undermine them, create doubt, create certainty, destroy the certainty, add characters, conflict, irresolution, resolution, chaos, and, then, order. Every class should have an answer to the question: What is this class about? Help them answer this question.

**But, I can’t throw out the national standards.**

Yeah, I know. This is where Standard Based Grading comes to the rescue. You see, SBG is a panacea…

**Hold on a sec…**

Wait! I’m on a roll here. Set your own standards, find the most important things you could teach a student. Teach it; teach it well. And then when they come and ask why your students don’t know about topic *xyz*, tell them, “I forgot.”

**???**

It’s Steve Martin, damn it.

**???**

All I’m saying is that once we teach our students the core ideas about how to think about the world in a mathematical way (read “logically creative way” not “creatively logical way”), then and only then does it make sense to start adding things back in. The standards are the problem. But you have to teach them the standards. But the standards are making it harder to teach and harder to learn. But you have to teach them the standards. But teaching them the standards is, in most cases, tantamount to making sure they will not learn any real mathematics. But you have to teach them the standards. I give up. Teach the standards. The story sucks, but so do most movies.

# Dénouement

I apologize for wasting your time. I don’t have any practical ideas. I’m lucky: I can get away, in many of my classes, with doing my own thing. If you’re teaching middle or high school, you are accountable to so many people that trying something like what I’m suggesting is probably career suicide. Don’t do it. If you’re reading this, you’re already too good a teacher to waste, especially on this rubbish. Maybe I’ll get back to reading Dan Meyer instead of trying to write like Dan Meyer.

Man, I started this post feeling so enlightened. Now, I feel…defeated.

Enjoyed this a lot. But, of course, I’m going to zero in on the one part with which I had a major disagreement:

If studying math has taught me anything, it is why struggle to learn things when others have struggled before?Why do people hike the Appalachian Trail? Don’t they understand that if they want to get from Georgia to Maine they can take a

car, or even aplane?There’s incredible value to be had in the experience of struggling with new concepts. Just as one example, if we wrestle with new problems enough, the outline of the puzzle piece we’re missing will come into sharp focus, at which point the new tool, when introduced, will be cherished, not simply added to the “list of stuff to memorize.” Isn’t that kind of Dan Meyer 101?

I realize your son is only 4. But if he were, say, 8, and playing Zelda for the first time, are you telling me you’d show him the walk-through right away? Talk about boring and pointless.

Sorry, I grew up on NES as well, and I guess I have some pretty strong feelings in that area.

Comment by Matt E — June 10, 2010 @ 9:30 pm |

Hi Matt,

Actually, I completely agree with you. That line was a great example of my poor ability to be sarcastic. It is why in Part 2, I write, “Currently, we teach from a walk-through” and mean it quite derisively. Sorry for not being more clear on this point.

Oh, and when my son is 8, I will be damned if I let him look up a walk-through for his games. When I was a boy, we made maps by hand and had to figure things out for ourselves. Grumble grumble…where’s my Geritol?

Thanks for reading,

Adam

Comment by Adam Glesser — June 10, 2010 @ 9:40 pm |

Oops. Normally I’m quite fluent in Sarcasm. I should have caught that. 🙂

Comment by Matt E — June 11, 2010 @ 6:32 pm |

Hey that’s me! That one paragraph still echoes through my mind. I might need to tattoo it to my body like that Memento guy.

Two comments:

1. How did I not know there was a Mario Bros and Zelda cartoon? Wikipedia tells me that it came out right in my prime cartoon watching years. I blame my parents.

2. When you become a world famous blogger, I’m going to totally be the guy who was like, “I went to see Green Day when they were playing in a crappy karaoke bar in West New Castle.” Then I’m going to complain that you sold out. It’ll be awesome.

Really though you should credit Dan Meyer for having crazy blog traffic so even if I post something in the comments, that’s 1000s of eyes.

I’ll probably have something more insightful to say later but it’s like 12:30 at night and my brain isn’t fully firing.

Comment by Jason Buell — June 11, 2010 @ 1:23 am |

“When you become a world famous blogger, I’m going to totally be the guy who was like, “I went to see Green Day when they were playing in a crappy karaoke bar in West New Castle.” Then I’m going to complain that you sold out. It’ll be awesome.”

Sweet! Then I’m going to sue you for illegally sharing my blog posts on P2P networks. See you in court, bro.

Comment by Adam Glesser — June 11, 2010 @ 8:27 am |

…teachers also need to be aware of the overall story of mathematical education along with the day-to-day practice of teaching concepts, methods and algorithms.I couldn’t agree with you more, and somewhere I have this beautiful vision of a year in which math stories are told not just day-to-day, but as a part of some grander story as well– but before I can get carried away imagining how thrilling and empowering it would be for our students and teachers to understand math in that way, I realize that we have standards and curricula and textbooks and high-stakes tests that aren’t strung together quite so majestically.

Even if we were to ignore those, I suspect it takes a much higher level of mathematical prowess (and storytelling prowess) than many of our brand-spanking-new teachers enter the classroom with to see the connections, recognize the drama, and construct a curriculum like that. Plus, they’ve got all the beginning-teacher technical skills to master, like classroom management and grading.

This raises two questions for me: is it reasonable to expect that our teachers will start with a more walk-through approach and graduate to a more storytelling approach? And to what extent would teachers need to write the story themselves to internalize and teach it well– or could they adopt it from a really well-written guide?

Thanks for the link/shout-out, and I look forward to more conversation!

Comment by grace — June 11, 2010 @ 7:18 am |

Hi Grace,

Thanks for stopping by and for the ridiculously provocative comments. Let me throw out some opinions and see where that leads us.

“I suspect it takes a much higher level of mathematical prowess (and storytelling prowess) than many of our brand-spanking-new teachers enter the classroom with to see the connections, recognize the drama, and construct a curriculum like that. Plus, they’ve got all the beginning-teacher technical skills to master, like classroom management and grading.”

You bet. There is a vicious circle here, though. Teachers are taught in the system and, naturally, most of them become system teachers. Even if there were no issues with classroom management, I still think we would have similar issues, but I think, to a large extent, the system creates those technical problems (would this be called “overhead”?). If math was taught as the beautiful, creative and practical discipline that it could be, I suspect our new teachers wouldn’t have to worry so much in the beginning about the technical skills of classroom management and grading.

“…is it reasonable to expect that our teachers will start with a more walk-through approach and graduate to a more storytelling approach?”

For most, I don’t think so. Anecdotally, once a teacher gets out of their first few years of teaching, very little professional development takes place (although, I admit, my experience on this point is mostly with college professors; perhaps K-12 teachers work harder at it than we do).

“And to what extent would teachers need to write the story themselves to internalize and teach it well– or could they adopt it from a really well-written guide?”

I think these things need to be modeled over a long period of time; I have little confidence that a training manual could do the job. A well-written guide might be a good resource for ideas, but it also might suggest that there is but one story (or one way of telling the story). As human beings, I think we all tend to look for “the way” to do something—it is certainly much less taxing on our brains if we’ve found it—but, usually, this just ends of being an excuse for not having to think any more.

Wow. Great stuff, Grace. I don’t think I’m done thinking about what you wrote.

Comment by Adam Glesser — June 11, 2010 @ 8:25 am |

One big idea that I find useful is “The Line”. Lines are easy. Students can solve linear equations. So much of the difficult and useless stuff comes from trying to give students the skills for calculus without the motivation for calculus, which is to try and treat functions like lines. Factoring is simply breaking things up into linear parts. Matrices are an efficient way to deal with lines. Probably 50-75% of what I teach I try to take back to lines.

Comment by Jason Stein — June 11, 2010 @ 10:23 am |

That’s a great point, Jason. I hadn’t thought about it so clearly before.

Comment by Adam Glesser — June 11, 2010 @ 10:27 am |

[…] of the Trade — Adam Glesser @ 5:06 am Tags: factoring After the up tick in hits from my last post, WordPress asked if I’d post something to decrease the load on their servers and so I bring […]

Pingback by Stupid Factoring Trick « GL(s,R) — June 12, 2010 @ 5:07 am |

“This is what our math curriculum feels like sometimes: a seemingly random ordered sequence of topics with often arbitrary repetitions…”

Answer? SBG? Or a coherent curriculum where one topic leads to the next? SBG just adapts to the lousy standards, without addressing the underlying problem.

Jonathan

Comment by jd2718 — June 13, 2010 @ 4:16 am |