The Noon Day Project - Let the Measurements begin!

See the previous post (from last March) introducing the Noon Day Project.

Today (9/22/10) is the perfect day for doing the CIESE sponsored Noon Day project measurements. Why, you ask? Well, assuming you have lots of sunshine (like I do today in White Plains, NY) it's a great way to celebrate the Fall Equinox. This activity is a recreation of the famous experiment that Eratosthenes did over 2200 years ago to determine very accurately the circumference of the earth.

Want to learn more about the experiment quickly? Watch (skip Ad) Carl Sagan's 6:42 min story about Eratosthenes's amazing discovery 2200 years ago. And it's not too late. You can still sign up for the fall, 2010 running of the project. (See current list of participants from all over the world.)

It will be done again in March, 2011. Let me know if you want a "heads up" early next year.

References

Noon Day Central - http://ciese.org/noonday
Article: Ihor Charischak. In the Spirit of Eratosthenes - Measuring the Circumference of the Earth (Learning and Leading with Technology, ISTE. April, 1997) - pdf
Article: Al Rodgers. Eratosthenes in the Schools. - link

Next time: CLIME Update - Planning for the NCTM meeting in Indianapolis - April 13-16th, 2011

Is there a natural path to formulas?

Can there always be a natural path from the concrete to the formal for all abstract math ideas? This is the question that Dan Meyer was pondering during the Math 2.0 Elluminate session on 8/25/10 so he challenged his audience to help him to find a natural path to solve a perplexing mathematical object (PMO) namely "How do you turn the “rise over run” (counting units) method of finding slope of a line into the more formal/abstract slope formula: Y2-Y1 / X2-X1? (You can hear/watch Dan's 4 minute challenge below.)

In other words, once your students get a handle on finding slope from the graph, how do you get the students to use the formula exclusively? 

Dan said that his students will stick to the “lower level” skill of counting squares on grid paper to determine the rise over run and resist using a more efficient method where they can just plug in numbers into a formula. For example, here is a typical problem which expects you to use the slope formula.

Find the slope of a line that passes through point A:(-4, 2) and B:(8, -3).

Figure 2

The more intuitive and concrete approach is to plot the points and draw the line first, then count the units for rise over run and get the slope that way. But Dan wants his students to use the x and y coordinate values of A and B and substitute them into the slope formula to get  2-(-3)/(-4-8)  equals -5/12 answer. How do you motivate that?

Dan answered his own question in response to this question posed by Colin: "Do any of your students start asking you for the shortcut before they get involved in the problem?"  Dan said he would play "dumb" and see if his students can help him come up with with the formula. But this is only the first step. They still need training wheels until the idea of slope and the formal formula come together in a meaningful way. This can take a long time and students need to be reminded of the connection often before they can just fly with the formula. What is even more important for me is that if my students forget the formula they are able to recreate it from what they know about finding slopes from graphs. That means that the student really understand (or owns) that idea. That's all fine and dandy, but what if progress hasn't been made before the end of the class and the test is the next day? I would postpone the test, but if that's not possible. I would probably give away the formula since unlike the teacher in the comic above I don't have all day to wait for the eureka moment. There will be more teachable moments for me to try again. That's the best I can do. But in the long run I'm not optimistic that I can garner enough such moments to make a significant difference here. The challenge of finding all the natural paths is a tough haul for teachers if the overall trip is still the very scripted, but unnatural road to calculus. There is a better way to get there, but it requires a different way to look at things. We need to make he trip be intrinsically interesting.

Sure, knowing it from memory helps but for students who desperately seek a formula to memorize (like the girl in the comic) I would work on a better way to scaffold it so the student sees the connection between the concrete (the graph) and abstract formula. But this will always be an uphill unnatural effort. It requires something more than just a clever way to bridge the gap between the concrete and abstract. It needs a better context that gives students a reason to want to do it.

The most effective context I found to get kids to “own” this concept and facility with the formula was after playing several rounds of Green Globs. The students learned the necessary skills not because they had to, but because they wanted to. It was very useful for them to learn how to blow up more Globs with one linear function because they would get a higher point total and the reason that is so important more important is because in two weeks they would be competing in the Great Green Globs Contest. You see the more globs you can blow up with one line the more points you get. For example (figure 3) the function y=1.1x + 1 knocked out three globs for a total of 7 points. (1 +2 + 4).

Figure 3

With just a slight change in the slope (figure 4) an additional glob goes down and you get score of 15 (1 + 2 + 4 + 8) points instead. The value of the glob doubles for each additional glob that is hit with one line.

Figure 4

Understanding how Green Globs can inspire students to want to do math for its own sake is what makes finding the slope formula a very natural process and is at the heart of powerful learning.

Here's Guillermo teaching his

classmates about Globs.

Take what happened to Guillermo an unmotivated 8th grade math student who was introduced to Globs by his teacher. Using the slope formula was just the beginning for him. Once he discovered that you can also draw curves a whole new world opened up for him. Here is his report on how he handled an advanced Globs challenge.

At one point I asked Guillermo how many points he would get if hit all the globs with one function. He thought about it. Then I added "I think there is formula you could use." to which he responded "I don't need a formula. I'll figure it out." I thought he would then proceed to add 1 + 2 + 4 + 8 + ... + 4096 using a brute force method. But instead he surprised me with this a couple of days later. (Watch the video below.) Check out his score at the end.

Does Guillermo know the slope formula? I don't know for sure. But then who cares? He's miles ahead of that now. Math will never be the same for him. He will learn what he needs to know when he needs to know it. And that's what Math 2.0 is really all about.

Notes

Keith Devlin slide 1

*George Lakoff and Rafael Nunez (Where Mathematics Comes From?) believe that all math can be learned intuitively even calculus if it is taught in way that connects previous learning with current new knowledge. Keith Devlin doesn’t think that’s possible for subjects like calculus. During his opening session presentation** at the NCTM conference in 2004, he said that there is some math that just needs to be learned “top down.”  He poses this challenge to the audience.  

Keith Devlin slide 2

More about Globs

• David Kibbey one of the original co-authors of Green Globs continues to update the software and makes it available at http://greenglobs.net
• See my 3:40 min Globs intro video. 
• Read my article about "What you always wanted to know about Green Globs but were afraid to Ask" and 
• Glance through the Great Green Globs Contest website.

** You will need to download Realplayer to watch Devlin's keynote opening session. His keynote begins at 18:15 in the video. He starts talking about the Lakoff and Nunez book at 46:45.

Unnatural Currents & WWCDAI (What we can do about it) Part 1

Figure 1

Turning word problems into meaningful, natural experiences is one way of describing what Dan Meyer was showing us at last night's (Aug 25) Math 2.0 Elluminate session. He's developing prototypes for curriculum units that he is beta testing with the help of lots of enthusiastic teachers who are willing to go the extra mile to do something similar. What Dan's cadre of early adopters (many of whom attended the session last night) are doing is testing the waters for a new kind of curriculum called WCYDWT (what can you do with this) which asks teachers to be creative designers of learning experiences for their students. For example, what would you do with the picture of a dog wearing a "triangular" bandana (figure 1)? Certainly NOT what the authors of this textbook example did. But the truth is as Dan pointed out in the session these kinds of contrived examples - unnatural currents, as he calls them – are difficult for students to own as real.

Figure 2

Classic math word problems tend to suffer from this contrivance affliction. For example, here's one (see figure 2) that Dan shared with us. From my experience this is a hard problem for most students.*  I still remember them from my own high school days and the only way I was able to do them was to follow the recipe that was in the book. And since I was pretty good at following recipes without making careless mistakes, I usually got A's. Was I good at math? Conventional wisdom said yes.  But the reality was that I was burying a deep, dark secret. I had no idea how to solve the problem without major scaffolding support. Without that net I would freeze and crash. And worse, I didn't really care, as long as I got good grades. Of course, that came back to haunt me in college when as a math major I could not hide behind my “Emperor's New (old?) Clothes." Luckily for me things turned out because (1) I discovered the 500 section of my college’s library and (2) later as a teacher, I became passionate about how best to teach math to kids. I taught Algebra I from a conventional textbook where these river problems did need to be crossed. I thought that if I demonstrated the logic of the steps in clear and concise language and if the kids understood each step that they would appreciate the power of math. Unfortunately, I forgot the lessons I learned from my own learning. Logic doesn't necessarily lead one to a profound understanding of math especially if the problem is perceived as boring. Dan's inspiration was to turn this contrived problem into a “real” one by substituting a moving escalator for the river.  The problem became how long does it take Dan to travel “up the down escalator.” (BTW – substitute staircase for escalator and you have the title of a cool education-themed book/movie from the 60’s.)

By solving the escalator problem (see Dan’s blog) he and his collaborators were doing a far more interesting albeit more challenging problem. The contrived version (figure 2) remains in the textbook still to be conquered or ignored as many of you might consider doing until you realize that not only might this problem be on one of those dreaded standardized tests but it's also a part of the way that Algebra works its magic and is very interesting to folks who like math. And if I want kids to appreciate math don’t we want them to be able to understand how the powerful ideas in Algebra makes solving problems so elegant?

Sidebar: It turns out the escalator problem may not be the best way to motivate the algebraic approach. (See Frank, Rob, Dan and Colin’s responses.)

I’ve started to think about a way to rescue our disparaged kayak problem and I think I made some progress. What I like to do is give my students a challenge (containing some perplexity** and fun) that’s in their zone of interest and skill level to pursue using manipulatives (virtual or physical) to help them concretize the formal language that will be used later when discussion about algebraic solutions comes up. (I like to think of formulas or rules as a shortcut way of solving the problem. But the caveat is they should only use formulas after they understand how and why they work. Part of mastery is being able to derive a formula when memory fails you.)

So I started looking for a manipulative to use with the Kayak problem. After a quick Google search I found a virtual manipulative*** written in Geogebra. I used the data I collected from this simulation and a spreadsheet to help me solve this in a way that I think would make sense to kids.

This is my initial collection of data. From my chart I see that that sum of the speed of the boat and the current is the same as the distance traveled divided by the time it took.

In the kayak problem since we know that distance = rate x time and distance = 12, and time upstream = 3 hours then the average speed of the kayak is 4 mph. Going down stream the kayak takes only 2 hours, so it's traveling 6 mph.

The only current stream speeds that are possible for the times to travel both up and down stream in the same time is either 5 or 1. Why? If stream speed is 5, then the kayaker could cover the distance downstream in 2 hours with a boat speed of 1 mph and vice versa – switch the 5 with the 1. The same works for upstream. This time subtracting the current speed  from the kayak speed has to equal 4. And 5 and 1 do that as well.

So in review, is this a contrived problem? Yes, indeed. Do we need better ways to teach it. Very definitely. The way to do it is for teachers to put on their creative caps and work with colleagues who can bring something to contribute to the WCYDWT curriculum party! And that’s what Dan is doing at his blog site right now.

Next time (Part 2): a powerful idea for connecting the abstract with the concrete. Inspired by Dan’s discussion of motivating students to move past counting rise and run from a graph with the more abstract short-cut:  y₂-y₁ / x₂-x₁

*My “most students” theory comes from my experience of teaching algebra word problems to freshman high schoolers. I didn’t say all because there were individual and classes of students that loved them and treated them like puzzles. As a student I personally wasn’t very fond of them because they weren’t the right kind of hard for me or in other words out of my zone, but I did them conscientiously so I could do well on the test. Sigh…
**Thanks, Dan, for the cool word.
***Online applet works best using Firefox (Mac users)

Math 2.0 Live! Wed August 25th 2010 9:30 EDT

Dan Meyer will be the guest host for the tomorrow night's Math 2.0 Elluminate session. See Events page. More about this event in my next blog.

The Madison Project, Bob Davis and the Mathman

"How can I motivate my students to get more interested in doing math?" was the question I posed to Don Cohen back in 1972 at a Saturday morning math workshop in NYC.  "The problem is that your kids are not really doing math," Don replied as we strolled down a picturesque Greenwich Village street. "What you need to do is get your students to create their own math. But first the teacher needs to do the same. That's the purpose of the workshop I am leading here." That one comment has stayed with me ever since as I continue my effort to inspire teachers to aim for that vision for themselves and with their students.

At the time, Don's vision came from his work with Bob Davis and the Madison Math Project. Join us tonight at the Math 2.0 Elluminate Live! session (August 4, 2010 9:30pm EDT) where Don (the Mathman) will be sharing that his early vision hasn't really changed all that much over the past 40+ years.

If you can't make it tonight, there will be a recording available at Math 2.0 Elluminate site after the session is over.

Patrick Vennebush - NCTM's Technology Linchpin

When Seth Godin wrote his book Linchpin, he had people like Patrick Vannebush in mind. Last night (July 7th) Patrick was the guest speaker on Maria Droujkova's Math 2.0 Live! Elluminate session and he spoke very eloquently and passionately about his work as one of the technology leaders at NCTM. 

As people who read my blog know, I have this "love/hate" relationship with NCTM. On the one hand I'm dismayed by NCTM's continued conservative stance when it comes to promoting their technology principle while their support of the other 5 principles (equity, learning, teaching, assessing & curriculum) is loud and clear. Since I'm all for equal time for technology and I wanted to do a non-confrontational protest of some recent decisions that the NCTM board made about the use of technology at conferences (i.e. vetoed support for using Wifi at their annual "showcase" conference because it costs too much) thus discouraging the use of a technology that is currently ubiquitious in society. Instead at NCTM meetings where you still have to line up at their anachronistic "Cyber Cafe" which worked in the late 1990s-and early 2000s  when the Web 2.0 was still emerging. Since NCTM provides discounts for affilate groups (of which CLIME is one) I was able to exhibit and promote Math 2.0 which is a synergy between the use of powerful math software tools in a Web 2.0 environment in math classes. Though it was fun sharing example how math ed will be transformed by these current and emerging technologies at the conference, it was discouraging to see how far we still have to go to educate our math teachers on its potential. 

Which brings me back to Patrick. Since the idea of CLIME began 23 years at an annual meeting in Washington with a vision promote a constructivist, student empowered way of learning with technology,i've been involved with NCTM as a affiliate president, conference planning committee member  at the national and local level (1999 and 2004) and on several committees, I have met a lot of people intimately involved with NCTM and the surprise was that weren't any conservative folks who were trying to keep us in the 20th century, but were truly people who were interested in reform with technology being at the forefront.

So NCTM by its a nature is conservative, yet at the same time is encouraging and doing some creative & innovative work. Patrick is one of the points of light that has emerged on the NCTM scene that bodes very well for the future and the direction that NCTM is headed. 

Read more about Calculation Nation and the Illumination project at NCTM.

Hear Patrick's archived presentation on Elluminate July 7th, 2010.

WCYDWT defined? more...

Here's another "What Can You do With This?" lesson grabber. Watch the video. What are your reactions?
My reaction was (and still is): "This is cool!" My next thought was: "Can I turn this into a math session motivator?" That is, can I get my students attention long enough to transition into an engaging & productive math activity? (As always, there is a devil ready to snarl progress in developing the details.) Dan Meyer who shared this video on his blog is thinking about how this might become a starting point for building a curriculum unit around it. I like the idea a lot because it feeds my mantra of what pieces a lesson scenario should have.

I say "we" in step 3 because as the teacher I'm learning right along with the kids so I want to share as well.
Dan has gone further by creating a more story-like 3-step process rubric. From his latest blog entry:

(In the) Beginning:

• engage the students with multimedia — pictures, videos, sound.
• the students come up with the question.
• the students make predictions — “give me a guess.”
• the students establish a range around their answer — “give me a wrong answer. give me an answer that’s too high, that’s too low.”
• there isn't information on the first image.
• “announce the problem’s constraints quickly and clearly.”
• ask questions that lend themselves to guesses: "how long? how many? how heavy? how far? how fast?"
• try to translate questions that are harder to guess into questions that don't change the objective but which lend themselves to guesses: "what is the area? what is the circumference?"

Middle:

• ask: “what information do you need to solve this?”
• ask: “how do you know that?”
• ask: “why?” “how?” — even on right answers.
• encourage students to explain their reasoning to other students.
• ask students to collaborate — “what do you think about jerold did?”
• ask: “how would that help you?” after they tell you certain information is necessary.
• ask: “what isn’t necessary to find the answer? what information don’t we care about?”

End:

• ask students to summarize the process.

In my "model" the middle portion (above) is still in the setting the stage phase, but that doesn't detract from these suggested steps being very helpful in framing my actions in a way that maximizes the possibility for learning.

Not all "stage setter" learning objects have the same impact. Dan's Water tank activity left me a bit dry, because I didn't find the video had much intrinsic motivation. (In other words, I didn't think it was "cool".) But like all devices for grabbing a student's attention, the context in which the learning object is presented is crucial. That's why I like to use video clips that pass my "Who cares?" test. It makes the transition to building the context around it smoother & easier.

See also my companion blog "Scenes from a Dynamic Math Classroom" for some of my WCYDWT tinkering.