I have the week off for Thanksgiving, so some friends and I have spent the last few days in Napa hanging out and doing some wine tasting. We got a recommendation to visit Freemark Abbey Winery, which we did, where I came across what seems like a pretty good problem-based investigation. Here is the scenario as they describe it (based on actual events):
"In September 1976 William Jaeger, a member of the partnership that owned Freemark Abbey Winery, had to make a decision: should he harvest the Riesling grapes immediately or leave them on the vines despite the approaching storm? A storm just before the harvest is usually detrimental, often ruining the crop. A warm, light rain, however, will sometimes cause a beneficial mold, botrytis cinerea, to form on the grape skins. The result is a luscious, complex sweet wine, highly valued by connoisseurs."
Basically, they had to decide: harvest the grapes now and guarantee the production of a moderately priced wine, OR wait for the incoming storm and hope that it encourages the mold growth necessary for production of high end wine. There are some accompanying details that the winery used in the decision making process (which I have somewhat shortened here):
  • There was a 50% chance the storm would hit Napa Valley
  • There was a 40% chance that, if the storm did strike, it would lead to the development of botrytis mold
  • If Jaeger pulled grapes before the storm, he could produce wine that would sell for $2.85 per bottle
  • If he didn't pull and the storm DID strike (but didn't produce the mold), he could produce wine that would sell for $2.00 per bottle
  • If he didn't pull and the storm DID strike (and did produce the mold), he could produce wine that would sell for $8.00 per bottle (although at 30% less volume because of the process to produce this wine)
  • If he didn't pull and the storm DIDN'T strike, he could produce wine that would sell for $3.50 per bottle

I can imagine several different versions of the problem depending on the group of students you were working with (by adding/removing details and alternative scenarios). I am attaching the printouts that I photographed in case anyone wants the details.
File Size: 2879 kb
File Type: jpg
Download File

File Size: 2842 kb
File Type: jpg
Download File

I recently asked a student of mine the question:

Do students care about 'real world' connections and problems or is there something else that motivates and drives them?

Her response:

"To be perfectly honest, as a student I do not know really know what I need for the real world. The term real world confuses me sometimes. While in the process of learning something I have never thought to myself "This will really help me later on in life. Especially while I work on so and so in the future" because it's irrelevant. The helpfulness for the "real world" comes later and I think that we can never really realize the significance until we naturally and effortlessly apply that knowledge throughout our daily lives.

When I am bored or I don't want to try or the assignments seems super difficult and tedious, I subconsciously think to myself "all this effort will contribute to nothing for me later on in life. When will I ever need this? Never." And therefor justify my actions for either not doing it or not caring or not doing the work to my best ability. 

extreme challenges with no personal connection or interest, or very confusing requirements = I won't need this!

I believe that most student's are not motivated by the real world. What students want is something that they can personally connect with on a deeper more intimate level. Something exciting, current, and easily relatable. 

I think relatable can translate to most students definition of "real world." When we argue for something real world and applicable, I think what we really mean is that we want something relatable and personal." 

Students have a lot of interesting things to say about education. Maybe we just don't ask them for their opinion often enough?
Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem.  I would love some blogger critique so please let me know what you think...

HORIZON: How far away is the horizon?

This is a pretty famous problem in mathematics (I think). I remember seeing it and being intrigued. I think there is an opportunity for pretty rich math here. Students would have to create an abstraction to represent the situation, develop rules about their abstraction, and use them to approximate an answer to the unit question.

PROBLEMATIC PACKAGING: How can you optimize this packaging?

This one arose out of a curiosity I had about optimizing parking lot design. I thought it might be fun as a puzzle where students get a package and different "items" (blocks of two or three sizes and shapes and worth various points) and they have to figure out how to maximize their point value. Could be extended by looking at different point values or different size boxes.

GET ME OUTTA HERE: How do you know if a game is solvable?

I love these puzzles. I'm not exactly sure how to turn this into a full-blown unit, but I'm pretty sure it can be done. This might not be the best unit question. I have also considered giving students a specific puzzle and asking "What is the fewest number of moves?" Maybe we can extend it from there and move into "solvable" setups?

7 CLICKS FROM KEVIN BACON: What is the minimum number of clicks to get to Kevin Bacon from ANY person on Facebook?

This one I'm REALLY not sure about. Obviously, it comes from the popular game about social connection (although it might be a good idea to use a celebrity my student will have actually hear of). I thought there might be some connections to graph theory here and it might be a nice extension of a combinatorics unit that I will do again next year ("How many combinations are there at Chipotle?"). Feedback please.

WIN, LOSE, or DRAW: Can you draw this without lifting your pencil or going over the same line twice?

The Bridges of Konigsberg is such a GREAT problem that I want to turn it into a whole unit next year. I'm not sure I love the context of the original problem because, to students, I think it feels like a pseudocontext. I thought about giving them a crazy network diagram and asking the unit question about that (maybe with "bridges" showing up along the way?). However, I usually prefer to start with a concrete situation and abstract from there....not sure.

...something more complicated but this was the best picture I could find for now

GOING COASTAL!: How long is the California coastline?

I did a similar project this year where students figured out the area of the Koch Snowflake. I'm thinking about trying to start concrete and move abstract next year with this one. I'm sure the Koch Snowflake will still rear it's ugly head somewhere in our investigation.
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 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.
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?!?"