Penny Circles

This discussion represents the final Makeover Monday Problem of the Year. I must admit, I found this problem to be easily the most engaging for me simply as a curious math learner. (Here’s  the original problem posted by Mr. Meyer.)

Now, I will admit that this is going to be more of a discussion of my experiences exploring this problem and, if you get lucky, I might come up with a recommendation somewhere toward the end.

The problem presents a rather interesting set up: filling up circles with pennies, making predictions, modeling with data, best fit functions. Lots of different entry points and possible rabbit holes to get lost in.

First change was that my circles were not created by radii of increasing inches. For the record, I can’t quite explain why I chose to make that change. Instead, I used penny-widths. Circle one had a radius of one penny-width. Circle two’s radius of two penny-widths. Circle three’s radius… you get the idea.

It looked like this:

I started with 4 circles with radii based on penny width.

I started with 4 circles with radii based on penny width.

Note: I understand the circles look sloppy. I had wonderful, compass-drawn circles ready to go and my camera wasn’t playing nice with the pencil lines. So, I chose to trace them with a Sharpie freehand, which… yeah.

Next, I kept the part of the problem that included filling each circle up with pennies.

It looked like this:

Penny Area 1

Then I filled them with as many pennies as I could

Then I filled them with as many pennies as I could

Here’s where the fun began.

I saw a handful of different interesting patterns that were taking place. If we stay with the theme of the original problem, then we’d be comparing the radius of each circle to the penny capacity, which would be fairly satisfying for me as an instructor watching the students decide how to model, predict, and then deciding whether or not it was worth it to to build circles 5, 10, or 20, or if they can develop a way to be sure of their predictions without constructing it.

But, I also noticed that by switching to the radius measured in penny-widths, the areas began to be covered by concentric penny-circles. So, we could discuss how predicting the number of pennies it would take to create the outside layer of pennies. Or we could use that relationship as a method for developing our explicit formula (should the students decide to do that).

So, if I am designing this activity, I feel like the original learning targets are too focused and specific. I want the students to be able to use inductive reasoning to predict. Simple as that. If they find that using a quadratic model is the most accurate, then cheers to them, but this problem has so much more to offer than a procedural I-do-you-do button pushing a TI-84.

First, I’d start with them building their circles using a compass and some pennies. Build the first four and fill them in. I’m quite certain that the whole experience of this problem changes if I leave the students to explore some photos of the my circles. They need to build their own. Plus, that presents a low entry point. Hard to get intimidated lining up pennies on a paper (although, I have plenty of student intimidated by compasses).

Next, I think that I would be if I simply asked them to tell me how many pennies they would need to build circle five and to “prove” their answer, I believe we would see a fair amount of inductive reasoning arguments that don’t all agree. The students battle it out to either consensus or stale-mate and then we build it and test. It should be said, though, that as they are building the first four and filling them in, I’d be all ears wandering about waiting for the students to make observations that can turn into hooks and discussion entry points.

Then I would drop circle ten on them and repeat the process. This is where the students might try to continue to use iterations if they can figure out a pattern. Some will model with the graphing calculators (in the style the original problem prefers). I’m not sure that I have a preference, although I would certainly be strategic in trying to get as many different methods discussed as possible.

Then, when they have come to an agreement on circle ten, I’d move to circle 20 and offer some kind of a reward if the class can agree on the right answer within an agreed margin of error. Then, we’d build it and test (not sure of the logistics of that, but I think it’s important).

All that having been said, I see an opportunity to engage this in a different manner by lining the circumference of the increasingly-bigger circles with pennies so that the circle was passing though the middle of the pennies and seeing how, as the circles got bigger, how the relationship of the number of pennies in the radius compared to the number of pennies in the circumference. I suspect it could become an interesting study on asymptotic lines.

It just seems like this particular situation has, as Dan put it, “lots to love” and “lots to chew on.”

The Essentials of Project-Based Learning

On Thursday, I will be facilitating a collaborative session for teachers on Project-Based Learning, which, to be fair, is a topic that I have never considered comprehensively. I’ve never been asked to define it. I’ve never been asked to explain it and, quite frankly, until the organizer of EdCamp Mid-Michigan approached me to facilitate the session, I never considered myself a practitioner of it. I just tried to design lessons that were engaging and built authentic, lasting learning.

But, this EdCamp facilitator, who’s name is Tara, has known me for quite a while. We did our undergrad together. We did our Masters work together. She’s been on an interview committee that almost hired me. She’s aware of my shtick. Maybe she wants me to facilitate this through the lens of how I work. If that’s the case, then this takes on an extra degree of difficulty because I still feel like I have a lot of development left to do before people start emulating my approach.

Besides that, my approach isn’t really that complicated. For any activity I consider offering to the students, I simply try to ask a couple of questions.

1. What can I do to maximize engagement? (I’ve seen plenty of ideas that I have had flop simply because the students aren’t drawn in.)

2. What can I do to facilitate collaboration? (The most effective use of class time that I ever see is when a group of students are effectively working together to solve a problem.)

3. What can I do to create a problem that will provide a variety of ways to solve it? (If there is only one real method to solve it, then as soon as one student gets it, the “collaboration” will become that student communicating “the way to solve it” to everyone else. My favorite evidence of this is when I hear a few “I don’t get how they did it, but this is how we did it and the answer came out pretty close to the same.”)

4. What can I do to make sure that the solution(s) are approaching effective learning outcomes? (While it can be interesting to occasionally have a group find a way to solve (or estimate a solution to) a problem effectively in ways that circumvent my desired learning outcomes, if I am continually putting together problems that don’t push the curriculum the community is trusting me to teach, then my students will be missing something.)

My favorite examples of this process working well are The Lake Superior Problem, The Pencil Sharpener Problem, The Wedding Cake Problem, and the Speedometer Problem. In each case, engagement and collaboration were high. Multiple solutions were present and most of the solution techniques required the students to make sense of mathematical procedures that could be correct or could be incorrect. “Why this formula is better than that other one.” Or “neither of our answers are perfect, this this one is better because this process has less of a margin for error.” Stuff like that. These conversations provide great opportunities to move beyond the memorization of a formula and to push toward sense-making of how that formula is used effectively.

While I’m not sure if this is “by-the-book” project-based learning, I have seen improvements in my students’ learning through the lesson design process I described above. Does anyone have any advice for me? Questions for me to think about? Corrections? Obvious holes in my logic? I appreciate this community because of your willingness to share, so feel free to do so.

Learning from Playing Around

These past two weeks have been an awesome time of learning for the students we’ve been working with, but I’ve also done a bit of learning myself.

I’d like to have my students love math and science and naturally be interested in it. But they’re kids. They would prefer to play. People get the most out of that which they put the most in to. If given the chance, students will put a ton into playing around.

These past two weeks I’ve been working with upper elementary-aged students. I normally teach high school students. I’m not sure if the age difference changes anything. The stuff they want to play with might be different, but not the desire to play.

And after 7 years of teaching math, there’s something appealing about a situation where students will be voluntary and enthusiastic participants.

Play Science 1

I have just spent two weeks watching students play with two activities. The first was an activity called “Table Timers” where they were challenged to design and construct an apparatus on a table top that reliably moved a marble down an inclined table in ten seconds. Second, The Helium Balloon Problem challenges students to keep a helium balloon rising, but to have it travel as slowly as possible upward. Not every group worked well and not every group achieved these goals, but the engagement level has been high. I suspect this is because they we allowed to play.

Here’s what caught my attention the most: In the midst of their play, the students demonstrated some authentic problem-solving techniques. They had to identify the major challenges to their goal, which they often did. They had to brainstorm possible ways to overcome the challenges, which usually took the form of raking through a tub of blocks or looking through the supply table. They discerned which seemed like the most realistic and then test. Following a test, they discussed what happened, why, and revised. And the students were often quite excited when they got the right answer (knowing themselves that it was right and not relying on me to tell them).

That’s a pretty good learning model. That’s something that I have a hard time getting my students to do with book work.

Play Science 2

So, the sharing, the idea-making, the consensus-building, the authentic assessment are all good things. Obviously I am not simply advocating letting the students play around all day. But perhaps by using play, we can improve engagement and the students seem to more naturally fall into a more authentic problem-solving mindset. When I consider helping them draw out the learning, some thoughts come to mind.

First, it seems like during the whole process of exploration, design, construction, testing, revising, and demonstrating, there needs to be an abundance of contents-specific vocabulary. The marble didn’t “bonk into that block.” That block “applied a force” to the marble. Students don’t “figure out how big the shape is.” The “find the area” or “circumference” or “volume.”

Second, students don’t seem naturally inclined to take data or to keep records. In the past two weeks, it seems that students are avid experimenters and do a pretty good job of verbally analyzing the problems if the plan didn’t work. Practically NONE of them documented anything on paper. No sketches, no data, no records of updates. This is an important part of the problem-solving process that would have to be established as a norm.

Third, the activities have to be tiered. Video games are great at this. The entry point tends to be quite low. The first couple of levels are pretty manageable and then the intensity and difficulty pick up. People get locked into video games through that model and people get unlocked quite quickly once the game has been beaten. Both Table Timers and The Helium Balloon Problem worked with this model. Then entry point was low for both activities and it was easy enough to begin to approach the goal, but perfecting the design and executing the plan took much more care. Then, once they hit ten seconds, we’d challenge them to add five seconds to their timer.

Fourth, I think that the groups need to be expected to summarize and present their work to each other and to field questions from the class. Class norms should allow for questioning of each other’s work and students can learn a lot about their own design, but also about the content when they know that they are going to have questions coming from their peers. Also, it would seem like this would encourage more thoughtful designs, too. Besides this, idea sharing gives the students an opportunity to look at other designs, integrate specific vocabulary into more regular use, and get the students comfortable with collaborating.


Play Science 3

I don’t think that playing around is the answer to everything, but I know that in my own experiences, it seems to be the forgotten learning model and if I’ve learned anything these past two weeks, it’s that an environment that produces enthusiastic student participation shouldn’t be ignored.

Motivating Learners

What could make a bunch of 10-year-olds do this in July?

What could make a bunch of 10-year-olds do this in July?

Today was the opening day of Kids’ College, which is a two-week, half-day science academy for going-to-be 5th and 6th graders at Michigan State University. This is my fifth year getting to lead as an instructor. That makes this my fifth opening day. Today I was struck by some observations that I hadn’t noticed before.

First, I’ll set up the situation. After an auditorium-style presentation with the whole group (there’s 18 instructors each with a group like mine) getting through the get-to-know-you info (lasting about an hour), we split up into individual groups. This year, I am leading 10 young people. I’ve never met any of them. Only two of them have met someone else in the group. After some introductions, we have about 75 minutes to use pipe insulation, tape, a marble, and anything else at arm’s reach to build a roller coaster that met some basic common guidelines.

It is of note (at least to me) that I wasn’t going to collect anything, I wasn’t going to record any grades, truth-be-told I wasn’t going to hold them accountable at all. So, here we go.

It wasn't quite right, so they fixed it. I didn't have to tell them.

It wasn’t quite right, so they fixed it. I didn’t have to tell them.

Observation #1: Off-task behavior was shockingly absent.  These kids had ideas, discussed them, and designed, reality-checked, and made predictions. I told them that I wasn’t handing them a marble until the coaster was built, stable, and ready for testing. My favorite question became “If this coaster has trouble, where do you think the trouble-spots will be?” Both teams had talked about it already. They knew. And they were both right, by the way. Also, no bathroom breaks. No kids asking to stop to eat their snacks. No kids asking what time it is and how much longer we have until we are done. I mean that. None.

There was no grade on the line. He simply didn't quit trying.

There was no grade on the line. He simply didn’t quit trying.

Observation #2: They decided their own “good enough”. I told them the requirements. They weren’t constantly seeking my approval. They didn’t ask questions like, “are we done yet?” On the contrary, in both cases, I had to instruct them to stop after fielding “aww, just one more test run? C’mon, we just need to fix this one hill… hold on.”

This team discussed, sketched, delegated without much direction.

This team discussed, sketched, delegated without much direction.

Observation #3: Students developed roles within the 5-person groups quite well. One guy was the tape guy. He held the tape, ripped a piece, applied it where the designer told him to. This person was working on the loop. That person on the curve. This person’s job was to tape the insulation to the wall to create the initial hill. These kids hadn’t ever met. I didn’t see any squabbling. I didn’t see any hurt feelings. I didn’t tell them to divvy up roles. About all I said, “You better get this organized!”

Okay, now I’m already arguing with myself:

“Yeah, but these are learners who are interested in science.” Okay, perhaps, but that doesn’t explain the group dynamics. Besides, putting these kids back on the busses to go back to meet their parents was all it took to reveal to all of use that these were definitely normal 10-year-olds.

“You work with 15- and 16-year-olds. This stuff must be easier with 10-year-olds.” Is that true? I have friends who teach elementary school and they gripe sometimes, too. I’m not sure that simply supplanting my typical group of 15- and 16-year-olds with 10-year-olds magically makes a lesson plan more likely to succeed. Does it? (A bit of help in the comments would be fantastic from those readers in the elementary ranks.)

“You’ve done this activity before and it doesn’t always go this well.” Very true. I have had students in the past that are a bit tougher to motivate. I have had groups with super-dominant leaders who try to monopolize everything. (In fact, this year, I tried to get around that by adding the structured timing… first 10 minutes discussion/sketching… then no marble until the whole design is constructed.) I suppose the student selection forces-that-be may have blessed me with a good mix, but I can’t help but feel like there is something more at work here.

Here is a stand-alone lesson. Didn’t do anything to assess prior knowledge and there wasn’t an assessment following. Nothing was turned in. Nothing was graded. Science isn’t built on that stuff. LEARNING isn’t built on that stuff. Learning is built on the stuff that I saw today. Discussing, sketching, questioning, building, testing, adjusting, asking why. Perhaps we allowed for those things to increase by eliminating questions like:

“Is this going to be on the test?”

“When is this due?”

“Do we all have to turn one in?”

“How many points is this worth?”

I mean, believe me, I understand the role of assessments and obviously I have a situation where I’m not being held accountable much either, which gives me a lot more flexibility. But what if that is the key? What if all the off-task behavior that is getting in the way is being caused by grades, tests, and other stuff? What if I got a window into authentic learning? What if today I saw a formula that worked? How can I integrate the lessons that I learned today into the September-to-June environment?

All I know is that today I saw something work. I want my classroom to work like that.

I don’t know. I guess I have more questions than answers. Perhaps that is why I am asking the questions to all of you. I know that you have more answers than me. I look forward to your perspective.

Reflecting on the Common Core…

photo credit: flickr user "Irargerich" - Used under Creative Commons

photo credit: flickr user “Irargerich” – Used under Creative Commons

A lot has been said about the Common Core State Standards in the last year. Some of it has been by me. Some has been by guys like Glenn Beck who is not a big fanMost (if not all) states have some sort of a “Stop Common Core” group. There is even a #stopcommoncore hashtag on Twitter that turns up quite a few results (although some use that hashtag as a means of highlighting objections in the arguments of CCSS opponents.)

The pub isn’t all negative. Some groups, The NEA among them, have come out in favor. Phil Valentine has some good things to say in support.

It is possible that both sides are probably overstating the impact that the CCSS will have. That being said, I will admit that I have some opinions on the CCSS. This year is our first year introducing a new geometry curriculum that we designed around the CCSS. I’ve written a few pieces before this one that have chronicled my journey through a CCSS-aligned geometry class. For example, I’ve documented that the CCSS places a greater emphasis on the use of specific vocabulary that I was used to in the past. I have also discussed (both here AND here) that the CCSS has present the idea of mathematical proof in a different light that I have found to be much more engaging to the students.

As I read the different articles that are being written, it seems like the beliefs about the inherent goodness or badness of CCSS has a lot to do with how you view the most beneficial actions of the teacher and the student in the process of learning. It’s about labels. Proponents call it “creativity” or “open-ended”. Opponents call it “wishy-washy” or “fuzzy”.

I suspect they are seeing and describing the same thing and disagreeing on whether or not those things are good or bad.

To illustrate this point further, a “Stop Common Core” website in Oregon posted a condemned CCSS math lesson because the students “must come to consensus on whether or not the answer is correct” and “convince others of their opinion on the matter.” The piece ends with “What do opinions and consensus have to do with math?”

The authors of this website are objecting to a teaching style. They are objecting to the value of a student’s opinion in the process of learning mathematics. Fair enough, but that was an argument long before the CCSS came around. I can remember heated discussions during my undergrad courses about the role of student opinion and discussion. (My personal favorite was the discussion as to when, if ever, 1/2 + 1/2 = 2/4 is actually a correct answer. One of my classmates rather vehemently ended his desire to be a math teacher that day.)

The CCSS have become a lightning rod for a ton of simmering arguments that haven’t been settled and aren’t new.

Consensus-building and opinions in mathematics vs. the authority of the instructor and the textbook. Classical literature vs. technical reading. The CCSS have woken up a lot of frustrations that are leading to some high-level decisions such as the Michigan State House of Representatives submitting a budget that blocks the Department of Education’s spending on the CCSS.

It is a little strange thinking that I am making a statement in a fairly-heated national debate every time I give my students some geometry to explore, but it seems like I do.

And I am prepared to make that statement more explicitly as I continue this reflection.

Designing Engagement and Collaboration

Collaboration and engagement are the goals. Learning comes through these.

Collaboration and engagement are the goals. Learning comes through these.

High school students can be capricious. Especially here in the nether regions of school-after-Memorial-Day. It can be hard to predict what type of assignment will engage the students. Something that has worked in years past might flop this year. But today, something worked.

Mind you, when I say that it “worked,” what I mean is:

A. It engaged more than 90% of the students.

B. It challenged more than 75% of the students.

C. Almost all of the students who were challenged persisted through the challenge.

D. It inspired honest and effective student-led collaboration.

That something is a handout that I like to call Wrap Battle.

(Now, fair warning: I didn’t write the handout and I don’t remember who did. I edited it a bit, that’s all. If you who wrote the handout are reading this and want your credit, I will be more than happy to provide it. Please let me know and I will make sure and give you all the credit.)

So, why did this handout contain work so well?

Well, first (and I feel like this is the most important), the situation in which the problem exists is easy to understand. Practically everyone has either wrapped a present or opened a wrapped present. So, there isn’t any students lost in the context of the problem.

Second, the actual number crunching isn’t overly complicated. This problem consists of adding, subtract, multiplying, and dividing of positive, whole numbers. You aren’t going to lose any students who have an idea of what to do, but get lost crunching the numbers.

Third, this problem has high predictability. The always important question, “Does your answer seem reasonable?” is a great turnaround when the students inevitably come to you with their paper and ask, “Is this right?”

Finally, that leaves us with the meat of the energy being designing the solution and testing/comparing results, which, at this point in the year, is exactly what they should be doing.

So, just a like a workout that is designed to isolate a certain muscle group, this problem approaches the students so that all of them can play, contribute and analyze the result. When they feel like they can, then that increases the chance that they will try.

What’s the Opposite of Success?

I want to explore two very important questions: How? and Why?

These questions tend are at the heart of the talk over how to “reform” America’s schools. Everyone from Sir Ken Robinson and Dan Carlin to Sal Khan and President Obama has an ideas. Some I agree with (the first two), some I’m not completely sold on (the last two). Either way, they all agree on this: something’s wrong and we need to fix it.

How do we go about fixing things? Bush’s No Child Left Behind, with it’s younger brother Race to the Top has something of a logical plan for creating success. Except that is isn’t really working very well. When it doesn’t work, then you have to move past the why-it-should-work-explanations and move toward a new set of discoveries.

What if there was a place where education is working well and we could explore HOW the system works. We can deal with WHY later. We have theorists and researchers who love to publish papers. They can work on the WHY. I am a practitioner. I am entertained and engaged by the WHY question, but find the answer to the HOW question more fulfilling.

One place has found success by doing the following things:

  • Narrative grades only until Grade 5
  • After that, teachers stress grade “as little as possible”
  • Not comparing schools or students by standardized testing
  • Teacher training programs resulting in each teacher having a master’s degree with the expectation that they will be “experts of their own work.”
  • Allow high levels of teacher autonomy by not mandating curriculum from the top down
  • Highly emphasizing “soft skills” like analysis, creativity, collaboration, and communication.

By doing this, they’ve created a national school system that is among the very best in the world.

That place: Finland.

Now, before I get labeled a “Finn-o-phile,” I want to state up front that I have no particular affinity for the Finnish culture (although, full-disclosure, I am partially of Finnish descent). I am focusing on Finland’s system because it is working better than ours.

To explain, I am going to let two articles most of the heavy lifting on this one. One from the Globe and Mail out of Toronto and the other from The Huffington Post.

I want to isolate some quotes from these articles:

The First from the Globe and Mail:

One of the ways the Finnish education system accomplishes [its success] is by giving individual teachers greater autonomy in teaching to the needs of their classes, rather than a top-down, test-based system.

America is currently moving away from this model. You can think good things or bad things about the content of Common Core, but the message is clear: Across the country, we want everyone doing the same thing. The Finnish system does have a National Core Curriculum which are defined  as “the legal norm for educational institutions” (sincere thanks to Dan Meyer (@ddmeyer) for fact-checking me on that) although discussions of assessment are much different than those of NCLB(which mandate statewide testing) and instead focus on assements “guiding and motivating” students as well as developing “their abilities in self-assessment.” (Quotes from the Finnish National Board of Education)

Also from the Globe and Mail:

The reality in Canada, which is unfortunate in Dr. Sahlberg’s view, is that students are rewarded for competing against their peers, teachers are held accountable by their class’s performance on exams, and schools are compared through widely published standardized test results. Finland takes an alternative approach.

The story is the same below the Canadian border as well. Standardized testing is THE evaluation tool for most schools and teachers. Real estate agents love it because it is so ingrained in our culture that parents will move into communities with good test scores because we’ve been conditioned to think that those are measures that tell the whole story. The Finnish system does the opposite.

Also from the Globe and Mail:

In addition to emphasizing collaborative work, Finnish schools have a different conception of knowledge than the traditional one. Teachers don’t think of knowledge as a cumulative store of objective information. “It is not primarily what individuals know or do not know, but more what are their skills in acquiring, utilizing, diffusing and creating knowledge that are important for economic progress and social change.”

Perhaps exposing a bit why standardized testing is avoided in Finland, these “soft” skills that are difficult to assess off a bubble sheet. According to Finnish National Board of Education, the National Core Curriculum includes options for on-the-job training with flexible assessments in which student can earn credits through “set of work assignments, a written paper, report, project assignment, product or equivalent” completed “performed individually, in a group or as a more extensive project.” American policy-makers are starting to appreciate these skills. Indeed, have you read about the Smarter Balanced Assessment? Leave it to us Americans to try to find a way to create such a standardized test.

From the Huffington Post (quoting Finland’s Minister of Education, Ms. Henna Virkkunen)

Our students spend less time in class than students in other OECD countries. We don’t think it helps students learn if they spend seven hours per day at school because they also need time for hobbies…

We seem to think that if students are struggling, they need more time in school. The Finnish system does the opposite.

So, let’s recap: Less time in school. Less testing. Less competition. More success. Could you imagine an American Politician standing on that platform?

The Finns have produced a system based on trust. They trust the teachers, they trust the local districts, they trust the students. The American system is based on a lack of trust. We call it accountability. We mandate curriculum because we don’t trust local districts. We over-rely on standardized tests because we don’t trust the teachers. We want longer school days because we don’t trust the students.

There is a nation that is excelling at education. They are, in many ways, doing the exact opposite of the things that we are doing. We, who are eagerly seeking to improve our system, are putting our hopes in standardized testing and state and federal manipulation of school districts through funding incentives. Perhaps it’s too early to state boldly that American reform efforts will fail, but we can say boldly that there are places where real excellence is happening and those people are moving in the opposite direction.

We could spend weeks arguing/discussing/explaining about WHY the Finnish system works. Don’t get me wrong, that is important. But, what matters most to me is this: It works. We could be doing what they do. We’re not… and it appears we won’t be for the foreseeable future.