There seems to be a lot of effort recently to teach math through "real world" contexts. You may scoff at this a bit because, as we all know, this is NOT a new argument. If you stop and think for a bit though, you might realize that this is still the "heart" of the reform movement in math. Project-based learning, interdisciplinary classes, and even "WCYDWT" (which have all helped to better education) have, at their core, the desire to show students that math really does exist outside of a classroom. To be clear, in many ways I think this is a HUGE improvement but, and maybe I'm asking too much here, I think we can do more than that (as I alluded to in a previous post).

I'm lucky to have a wonderful advisor, Stacey Callier, in my grad school program. I work in a project-based school and, as she knows well, I often push back against the hidden assumption in PBL that math is a "tool" that helps us solve real-world problems (I think the terminology I used today was that math seems to always be the "servant of science"). We started talking a lot about the constructivist philosophy that is central to my work and we (mostly she) came up with this matrix:
Is it better if we situate mathematics in a "real" context that students find engaging? Of course! I just think we can do that AND STILL honor a student's way of understanding and knowing, give them opportunities to author and create their own mathematics, and help them construct their own meaning for ideas that help them solve a problem. Call me an optimist (read: delusional), but I think its possible. 


I have been thinking about this a bit since I posted it yesterday and I'm not too sure how I feel about it still. There are a few things that are bothering me:

1. The "HOLY GRAIL" label implies that this is where math education "should" be…which I'm not sure everyone would agree with (in fact, I'm not sure I can say that this is where I think it should be.

2. The top (applied vs. non) seems to get at "why teach math" while the right (constructive vs. non) seems to get at "how teach math." Is it ok to compare these two things in a matrix? Not sure.

Mostly, I labeled the top left "HOLY GRAIL" because I strongly believe that math should be taught in a constructivist fashion. If we can do this AND situate the math in a "relevant/applied" context shouldn't we do that?!?
I'm sure I have explored/solved/assigned this problem before but I revisited it on a recent plane flight to Northern California. I thought I would just post the problem for now in case you want to give it a shot. I'll post my work tomorrow.

Which numbers can be written as the difference of two perfect squares?

p.s. As it turns out, doing this on a plane is also an excellent way to alienate the person sitting next to you. Apparently, Sudoku is totally acceptable but if you dare get any "math-ier" you are guaranteed to be met with shaming eyes when you look up (trust me).


Here is the work I did on this problem. I added some commentary in light blue typeface. I'm no number theorist and there are a few fuzzy uses of variables but I think the overall idea is pretty close...
I'm reading "Out of the Labyrinth" right now and even though I'm only about 50 pages in, there are some pretty powerful quotes. This one is my favorite:

"To teach it now as if it were A Rule, or (even more intimidating), The Law, is to pretend that what took years of experimenting and ingenuity is as obvious as your nose. And then, because you never really had a chance to understand what was going on, whenever you need this rule again it will come as just that - an arbitrary fiat, enforced by Them. And so the whole integrity of mathematics is compromised. The only reasonable conclusion for a struggling student to draw from such pretense is that he is irremediably stupid, or that Mathematics works in mysterious ways, its wonders to perform."

and further down the page:

"...and so a teacher, who is supposed to develop our powers of inquiry, becomes instead a messenger of Received Truth."

There are many other incredible little tidbits in this book and, so far, it is making a pretty good case for a spot in my preferred reading list. As I read the above passage, I was reminded of something that I have been thinking about a lot lately (and even alluded to at the end of this post). We talk a lot about the importance of context in math education. There are many benefits to situating mathematics into some nice context that students find engaging/relevant (mostly, it seems beneficial to help students see that math can help them describe and understand their world) and there are some sites/people that are doing this very well.

But simply using an exciting context does nothing to remedy the fact that often the "whole integrity of mathematics" can still be compromised. I mean, you can start with a really interesting and perplexing question and still completely miss the boat when it comes to helping students do their own mathematics; it just becomes a better way of teaching "The Law." These types of questions could (and should) drive a whole unit because then we can explore different parts of the problem, honor different approaches and ways of thinking, and ultimately, help students create their own math along the way.
I had grad school tonight and we watched this TED Talk from Benjamin Zander. To me, it was incredibly powerful. I'm not sure if that was because of where my head has been recently with thinking about education reform or not but, regardless, I think you will make your own connections to education. It is twenty minutes long and you might be wondering what it has to do with education but you HAVE to watch it until the end. For me the magic happened exactly at 17:15 and 18:23. I would love to hear what it meant to you.
There has been some talk here recently about the fact that most mathematics textbooks are trying to serve as both reference texts and instructional materials and that, inevitably, being the first almost guarantees that nothing of quality can happen in the second. In other words, if you are telling a student what/how to think (by offering a reference to someone else's mathematics) you can never have quality instruction (which, to me, would mean having students create their own mathematics). 

I would argue pretty intensely for taking any reference material out of students' hands (I mean, did you you miss the fact that the site was titled DOING mathematics?!?). I wonder, however, if there is space for helping teachers learn to use textbooks that already exist as a launching point for creating their own quality instructional materials? This, I hope, is the beginning of a post "series" about infusing "doing" into existing curriculum.

"Ok class, turn to page 900 (because there are way too many pages of reference and practice problems) and find the section on "Exponential Growth:"
This is the classic example of textbook as reference. It does everything short of telling you what a, b, y, and x represent (which came on the very next page). There is no opportunity for students to engage in creating mathematics here. But what if we change things just a bit. Have students watch this (live) and ask them this (or a variety of other questions...or have them ask one themselves):

We will be exploring global population until (insert date and time here).

I like this population clock because it doesn't give them any other information (births, deaths, etc) which allows us to generate that on our own. I also think there are more intriguing questions but I like this one because it allows them to make an initial estimate/calculation and then, ultimately, test the accuracy of THEIR model at the end of the unit/investigation (as opposed to asking "when will it hit 8 billion," which we won't see in our school year).

The point here is that NONE of the reference material is all! Students don't need a definition for exponential growth until they want one to verbalize what they are noticing about how population grows. They don't need variables and models until they introduce/create them themselves to describe the patters they see in that growth. They need to create this for themselves, not learn how someone else does it. The beauty here is that students will create their own models and we can discuss the benefits and pitfalls of each.

This simple redesign, or infusion, is a simple recreation of what already existed in a reference text. We can teach/trust teachers to create this experience for their classes without putting the reference text in the hands of every student (or trying to create a curriculum that is widely distributable). Of course, even with the "infusion" a lot of harm can still be done in how we help guide students in creating their own model...but that is a topic for a later post. 
Ask just about anybody to describe the typical math class and I imagine this is what you will get:

1. Teacher goes over answers from last night's homework
2. Teacher introduces new topic for the day while students take notes
3. Class does a few examples together
4. Students start the new problem set assigned

This format works if your goal is to have students learn (read: memorize) someone else's mathematics. If you want students to create, author, and do their own mathematics, something has to change. Brian Lawler introduced me to a lesson format that helps shift this focus:

1. LAUNCH: set up the task/question for the day
2. EXPLORE: students work collaboratively to explore
3. SUMMARIZE: students display work, ask each other questions, and make connections

More details about this format are available online. It's tempting to be a skeptic when it comes to this new format. Most of us have gone through a very different math experience and the pressure of standardized testing pushes us to cover as much "content" as possible. 

Our class is still breaking apart the "infinite star" problem and today's task was to try to figure out the total area at each iteration. The class came up with this:
…which resulted in this:
…and one student even wrote this:
We can talk all day about skills and testing but this sort of abstraction is at the heart of algebra and is the definition of what it means to "do mathematics."
I will often bring these rich tasks to our class for students and I have found that students not only enjoy them, but that these tasks have also been great for developing and reinforcing the "habits of a mathematician." I usually title these tasks "POWs" (for Problem of the Week). Yesterday, as part of the progression in figuring out the area of the "infinite star" (aka Koch Snowflake) I gave students this problem hoping that we could emphasize the importance of 'taking things apart and putting them back together':
I was pretty surprised by the difficulties students had with this problem. In fact, I had even included a more difficult one in anticipating students blowing right through this one. About 50% of the class was completely stumped by the original problem (which I appropriately named the 'Crooked House'…in case you haven't noticed, I appreciate a cheesy title). I asked them at the end of class to reflect on and discuss strategies that were helpful and even asked them why this problem was so difficult. They said things you might expect:

"we didn't have a formula"
"I needed the distance formula"
"I didn't know where to start"
"rusty geometry"

A little piece of my soul died. We have spent all year de-emphasizing formulas and talking about habits that give you a place to start. Then, this happened:

student: "It was like a POW, but not."
me: "What do you mean?"
student: "Well, it seemed like a POW-type problem but it wasn't one."
me: "So, just because it didn't say 'POW' at the top of the page you didn't approach it in the same way?"
student: "Yeah…kinda."

It was clear that students see math compartmentalized into separate worlds. I haven't helped them see that these habits ARE mathematics. They aren't just useful for puzzle-like problems, these habits are at the foundation of the creation and authoring of ALL mathematics. With my last two units I have started to include 1 or 2 of our habits as the central focus. Today's prompt looked like this:
It was a great reminder of the importance of redefining mathematics for students, helping them see connections, and making sure the message we send is clear and consistent in ALL that we do.
I'm pretty sure this wasn't Sam's intention when he asked, "how do you plan" but I wanna try this anyways. I have been doing a collaboration with students in the credential program at Cal State San Marcos where I gave them an idea for a problem-based unit and then they worked together to plan it. Working with them has made me think about: 1. effective structures for designing problem-based units and 2. how to help teachers plan units for themselves. I thought this might be interesting…

Here is the problem for the unit I started today:
My request is this:
1. Do the problem yourself
2. In the comments, describe how you would design a unit around this (big chunks, progression, etc)?

p.s. YES…I renamed it the "infinite star"….I realize the decision has pros and cons.
p.p.s. This problem launch is a classic example of how paper limits the mathematics. With my limited video skills, I couldn't do better.
There are a few themes in math education that are particularly interesting to me…habits of mind is one of them. They are the building blocks of "doing mathematics." The more I work with students, the more convinced I have become that they are also the strongest indicators of success in school mathematics. They are the things that good mathematicians of any age do on a regular basis. I seen a few collections of mathematical habits and there are certainly some great resources already (here, here, here, and here to start). After collecting and distilling, here is what I have so far:

Look for Patterns
Start Small
Be Systematic
Take Apart & Put Back Together
Conjecture and Test
Stay Organized
Describe and Articulate
Seek Why and Prove
Be Confident, Patient, and Persistent
Collaborate and Listen

I particularly like the last one because I get an amazing mental image of Vanilla Ice with his well-maintained flat top haircut and shiny "Hammer pants" ("STOP…collaborate and listen…Ice is back with a brand new invention…"). 
I have been toying with a feedback system for each that would look something like this:
Throughout the year we would keep track of how students progress in their development and implementation of each of the habits. Part of me hates to simplify and categorize the habits in this concrete way. The reality is that they are much more fluid and interconnected. At the same time, I think this helps students get specific feedback and helps them set specific goals for improvement. On that note, there are a few things I am wondering about and would like feedback on:

1. Is this a good list of habits? Are there ones that I am missing? Are there ones that are unnecessary? 
2. Suggestions for feedback systems? I hate to bring it up…but should this translate to a grade?
3. How can we rename the habits in a way that is more student friendly?


See my post here for an updated version of the habits.
This one has been bugging me and I really wish I had more readers because I think there is a valuable discussion here:

What is the perimeter?

How long is the coastline?

The perplexing question here is essentially the same. This guy talks a lot about the quickness with which we move to abstraction. I'm wondering if maybe that applies here. I like the abstract because, in terms of creating a unit with a nice resolution, it easily comes full circle. I'm not sure you can say the same about the coastline. So…"what's the difference?!?"