Modern stresses on classical systems

I was impressed by a lot of what I heard at #macul16 in Grand Rapids a few weeks back. (For those of you not in Michigan, the MACUL Conference is one of the biggest EdTech conferences in the Midwest. 4300 educators came together for three days of learning…)

But I can’t help but feeling like a wave has crashed on the shore. The overall messages sounded different than in previous years. Don’t get me wrong, there was still plenty of enthusiasm, but it wasn’t enthusiasm for technology proper. It seemed like the presenters were often asking: What kind of resources do we need to use to create the kind of cultures that will, as keynote speaker Rushton Hurley put it “make dynamic learning the norm?”

This was epitomized by Michael Medvinsky in his excellent talk about Culture of Thinking. The message was clear. FIRST decide what type of learning you want to see the students do. THEN decide what type of resources it will take to create an environment that is conducive to that type of learning. When the education system starts to process through those ideas and concepts, it can create stress in some very interesting areas.

What do we want the student learning about? Remember, we can talk about the TYPE of learning we want, but we still have to have some predetermined baseline for the WHAT of the learning. The maker movement and genius hour movement of recent years have inserted the importance of student-chosen learning time into the broader conversation, but the content we expect for EACH student by design says something about what we value as a culture. We need to take that message seriously. (I started hearing the term “passion-driven schools”. This actually makes me a little uncomfortable. More on that later.)

How do spaces like the library, the computer lab, and the cafeteria (before and after lunch time) play into our goals for our culture and environment? Come on a journey with Ann Smart and Kellie De Los Santos to see how the school library can be re-visioned. Also maybe Shannon McClintock Miller who models some fairly down-to-earth, but nonetheless super impressive redesigns for the position of librarian. Consider, too, that cafeterias tend to have tons of open space, high ceilings, varied structures for sitting, leaning, kneeling, doing work. And they are typically really, really empty during the school day with the exception of lunch times and overflow before and after school. In some places, this isn’t true as the cafeteria is also the gym, but in places where the cafeteria sits empty much of the day, what could be done with it that isn’t? What opportunities are being missed?

What’s the future hold for things like grades, calendars, credits, class rankings, GPAs, and so on and so on? The school environment is hanging on to a variety of structures that are throwbacks to a time when the industrial model of schools made a lot more sense. But modern changes are putting pressure on a lot of different things. Some of which are not getting discussed much in the conversations I’m hearing.

Case-in-point: I recently had a conversation with a teacher who had a variety of digital coding, IT, network security courses ready to roll out, free of charge, to students as young as eighth grade. He and I had a long conversation about how to get these courses in front of the students who’d be interested. And the primary sticking points? Well, first, the courses were typically projected between 40-60 hours to complete. That’s 8-12 weeks in most schools. (Schools normally work on 12-week, 18-week, and 36-week cycles). Second, what would the student receive at the end? A certificate from the course designer. (Schools usually operate in grades and credits.)

And I really feel like this isn’t small potatoes. (More on this later, too…) But this simple conversation about a perfectly reasonable idea did a great job bringing up how unprepared our structures are to cope with the flexible scheduling and grading practices that modern learning is going to increasingly require.

An idea like that? It changes the game. No cohort. No grade. No credit. And what do you do with them when it’s time for them to start something new in week 8 of a semester? (The same thing you do with the student who are ready two weeks earlier?) These absolutely aren’t insurmountable barriers. In fact, these are fairly solvable problems as long as schools are starting embracing a new vision for words like “course”, “learner”, “completion”, etc.

I got a what-if… What if we create a series of general elective courses that are designed in such a way that a student could enter at any time and be able to meaningfully join in. Maybe a phys ed, general art, theater, a project-based engineering course, and basic culinary. One for each hour of the day. Running both semesters. Courses with detached, independent units, something with a lot of DOING where the students who have been there longer are expected to model techniques for the new learners. Courses taught so that the successful completion of the course is judged based on the performance while there and not how long that time was. (That is, you can still earn an “A” for the semester having only been there two weeks. Even Our Lord realized how difficult that type of conversation can be.) It would be tricky. Especially at first. But not impossible. It will need to be designed intentionally, by people who are willing and able to do it well.

These questions have modern answers that bring with them a lot of potential stresses and unforeseen consequences. When you pile them all on top of each other, you just get a educational system that is ready to redesign itself from the foundations up. And I just hope that we’re ready for it. The proponents ready for the change not being as quick as they’d like and opponents ready to use their “yeah-buts” constructively.

Advertisements

We math teachers need to trust each other

I have had a great time working with 5th and 6th graders for two weeks this month. Kids College makes up some of the favorite weeks of my summer. I mean, after all, when a giant trebuchet is involved, it’s hard not to get excited.

But an interesting moment occurred when I looked over a student’s shoulder to see what was in their lab notebook. Here’s what I saw.

Student Work Big

See it? It’s significant… You know what? I’ll zoom in.

Student work

 

There we go! See that? That is a  straight-up attempt at long division. This might seem mundane and ordinary, but let me tell you why this grabbed my attention.

That student is trying long division. What does that mean?

That means that student was TAUGHT long division by someone. (While acknowledging that this student might be learning long division from a parent, tutor, pastor, or babysitter, I feel like the highest-percentage guess is that the “someone” is a teacher.)

This is important to me because I know several fifth grade teachers who have confirmed that it is common practice for each student to leave elementary school having had the opportunity to learn long division. (I promise, I have a point coming…)

Fast forward that fifth-grader about… mmm… 6 years. Now suppose they are learning this.

Polynomial long division

Taken from Holt Rinehart Winston’s Algebra II book, 2009 Edition, Page 423

This is, give or take, halfway through Algebra II. Now, I have seen firsthand that this isn’t the easiest skill for a lot of students to master, especially just as it is introduced. In fact, I would go so far as to say that there are plenty of student’s who successfully complete Algebra II while never mastering this particular skill.

There are a lot of reasons that students might not master this skill, one of which might be that THIS is typically the type of situational math problems that this skill gets applied to.

 

Applied Polynomial Division

Taken from Holt Rinehart Winston’s Algebra II, 2009 Edition, Page 426

But the one that I have heard mentioned to me with the most enthusiasm is that schools are starting to move away from teaching long division. And that division of polynomials is much more difficult to teach to students who haven’t been exposed to long division. Fair enough, except I have a couple of thoughts.

First, (and I’ll admit that this is a little off-topic) even if we assume that the struggling Algebra II students weren’t ever taught long division, what grade do you suspect they should have been? 4th or 5th grade, maybe? It just seems to me that any essential skill that was academically appropriate for 10-year-olds could very well be taught to 17-year-olds. I don’t see any reason to believe that long division is a skill with a window of opportunity to teach that is open to 10-year-olds, but has closed by the time students reach upper adolescence.

However, my second thought is that the evidence I gathered this week confirms what I suspect was true. They WERE taught long division. They fact that they can’t USE long division regularly in their junior year of high school requires a completely separate explanation. There are plenty of potential reasons why, but we shouldn’t allow ourselves to think that the explanation is as simple as “they were never taught that.”

This is a dangerous way to address academic deficiency. This has been a social complaint of schools for a while now. Jay Leno made a regular segment out of it. Every time a clerk has a hard time making change, or a young person appears to struggle balancing his or her checkbook, the question largely becomes “what the heck are they teaching these kids in schools?”

Well, I assure you all, that practically every American middle schooler has been in a math class that has covered the necessary skills to make change or balance a checkbook.

Just like we teach European geography, basic grammar, and the names of the Great Lakes.

But not every student learns it. And whose to blame?

I don’t know, but as a teacher, there aren’t a ton of folks giving me the benefit-of-the-doubt these days. We should, at least, be able to expect it from the person who teaches down the hall, down stairs, or in the building next door.

Common core is trying to deal with the “when to students get taught this” conundrum because there is a lot of social pressure that assumes that the problems with missing knowledge is missing instruction. I’m not sure if Common Core has what it takes to address that issue…

… especially if that isn’t the issue. Because what happens when we make sure that every fifth grader coast-to-coast is taught long division and 6 years later, coast-to-coast, the 17-year-olds are still unfamiliar with it?

Feeding The Elephant in the Room

I am going to ramble a bit in this piece, but as you read, keep a specific thought in your mind:

When our students have graduated high school, we will know we educators have done our job because _____________________.

Now, onto the ramble:

So, a lot gets said about the struggles of American secondary education. Recently, Dr. Laurence Steinberg took his turn in Slate coming right out in the title and calling high schools “disasters”. Which, as you can imagine, got some responses from the educational community.

Go ahead and give the article a read. I’ll admit that education is not known as the most provocative topic in the American mainstream, but Dr. Steinberg has written a piece that has been shared on Facebook a few thousand times and on twitter a few hundred more. It’s instigated some thoughtful blog responses. You have to respect his formula.

He starts with a nice mini Obama dig.

Makes a nice bold statement early (“American high schools, in particular, are a disaster.”)

Offers a “little-known” study early to establish a little authority.

Then hits the boring note and hits it hard. High school is boring. Lower level students feel like they don’t belong. Advanced students feel unchallenged. American schools is more boring than most other countries’ schools.

Then he goes on to discredit a variety of things education has tried to do over the last 50 (or so) years including: NCLB, Vouchers, Charters, Increased funding, lowering student-to-teacher ratio, lengthening the school day, lengthening the school year, pushing for college-readiness. I mean, with that list, there’s something for everyone

Like it or hate it, that is an article that is going to get read.

However, there isn’t a lot in the way of tangible solutions. The closest Dr. Steinberg comes is in this passage: ” Research on the determinants of success in adolescence and beyond has come to a similar conclusion: If we want our teenagers to thrive, we need to help them develop the non-cognitive traits it takes to complete a college degree—traits like determination, self-control, and grit. This means classes that really challenge students to work hard…”

Nothin’ to it, right? It’s as easy as making our students “grittier”.

Now, I will repeat the introductory thought: When our students have graduated high school, we will know we educators have done our job because _____________________.

That blank gets filled in a variety of ways from the area of employability, or social responsibility, or liberation and freedom, or social justice to a variety of other thises and thats that we use our high schools for. We are using our high schools as the training ground for the elimination of a wide variety of undesirable social things. We’ve used our schools to eliminate obesity, teen pregnancy and STIs, discrimination based on race, gender, or alternative lifestyles. We have allowed colleges to push college-readiness to make their job easier. We’ve allowed employers to push employability to make their jobs easier. The tech industry feels like we need more STEM. There’s a push-back from folks like Sir Ken Robinson who feel like it’s dangerous to disregard the arts.

And they all have valid points. I’m certainly not mocking or belittling any of those ideas.

However, very little is getting said on behalf of the school. We are treating the school as a transparent entity with none of its own roles and responsibilities. It is simply the clay that gets molded into whatever society decides it should be. Well, since the 60’s, society has had a darned hard time making up its mind about what it wants and so the school has become battered and bruised with all the different initiatives and plans, data sets, and reform operations. Reform is an interesting idea when the school hasn’t ever formally been formed in the first place.

So, we have this social institution that we send 100% of our teenagers to in some form or another and we don’t know what the heck its for. No wonder, as Dr. Steinberg puts it, “In America, high school is for socializing. It’s a convenient gathering place, where the really important activities are interrupted by all those annoying classes. For all but the very best American students—the ones in AP classes bound for the nation’s most selective colleges and universities—high school is tedious and unchallenging.”

Public enemy #1 needs to be the utter and complete lack of purpose in the high school system. We are running our young people through exercises… why? For what? What do we hope to have happen at the end? When we decide the answer to that question, then we eliminate the rest. It isn’t lazy to say, “I’m not doing that, because that isn’t my job.” It’s efficient. If you start doing the work of others, you stop doing your work as well.

We’ve never agreed on the work of the American high school, but I suspect some of what we are asking it to do should belong on the shoulders of something or someone else. I suspect as soon as we establish a purpose and simplify the operations around that purpose, we can start to see some progress on the goals that we have for our schools, which will spell success for our students and start to clean up the disaster that so many feel like the high schools currently are.

The 70-70 Trial

Education is a world with a whole lot of theories. Intuitive theories at that. I’m sure it’s like this in most professions. We are seeing an issue. We reason out what the problem seems to be. We determine what the solution to our supposed problem seems to be. And we implement.

The problem with that is problems often have multiple causes. Solutions are often biased. Results have a tendency to be counter intuitive. For example, a paper recently published suggested that the increase of homework might actually cause a decrease in independent thinking skills. This probably isn’t a conclusive study, but recognize the idea that if students aren’t demonstrating independent-thinking skills, prescribing a problem-for-problem course of study for them to do on their own might not be the best solution.

This leads me to a trial that I am running in my classroom for a semester. I have four sections of geometry. I am going to leave two as a “control group” (very imprecise usage, I’ll admit) that will run exactly the same as they did first semester. The other two will run “The 70-70 Trial.” This is one of those theories that has gotten tossed about our district many times. It seems intuitive. It seems like it addresses a persistent problem.

The theory goes like this: If you go into a test knowing that 70% of the students have 70% or more on all the formative assessments leading up to the summative assessment, then we know that the students are reasonably prepared to do well on the test. If you give a formative assessment, and you hit the 70-70 line or better, you move on with your unit, business as usual. If you miss the 70-70 line, you pause on the unit until enough of the class is ready to go.

This seems reasonable to some, and ridiculous to others. Our staff meetings have seen some pretty intense discussion over it. Proponents lean on the logic. How can a group of students with high scores on formative assessments struggle on summative assessments? Opponents speak to the time crunch. When do you decide to move on? You can’t just keep stopping and stopping forever? You’ll never get through the material. Both seem like logical points…

But, as far as I can tell, no one has tried it to see what would happen. So, I figured that I had two classes that really struggled their way through first semester. It became very hard to energize and motivate these students because of how difficult they found the material. Perhaps shaking up the classroom management and unit design will add a bit of a spark. These two classes will be the focus of the 70-70 trial. I will use this blog as a way to record my observations and entertain any ideas from people who are looking to give me ideas to help this idea work.

This starts one week from today. I don’t know if it will work. I have my guesses as to what I think will happen, but I am going to keep those to myself. I absolutely want to see this work because if it does, that means my students were successful. My chief area of concern is what to do when 61% of the students score 70% or better (for example). By the rules of the trial, I can’t go on. I need a reteach day, but over half the class finds themselves ready to move on. What do I do to extend the learning for those students, while supporting the learning for those who need some reteaching and another crack at the formative assessment?

These are the kinds of things I will be looking for help with. Thank you for being patient and willing to walk this path with me. I will look forward to hearing whatever ideas you have.

The Anxiety of Open-Ended Lesson Planning

It’s been about three years since I started weaning myself (and my students) off textbook-dependent geometry lesson-planning and toward something better. I’ll admit the lesson planning is more time-consuming (especially at the beginning), but most of the time expenditures are one-time expenses. Once you find your favorite resources, you bookmark them and there they are.

As we pushed away from the textbook, I noticed two things: First, the course became more enjoyable for the students. This had a lot to do with the fact that the classwork took on a noticeably different feel. Like getting a new pair of shoes, the old calluses and weak spots aren’t being irritated (at least not as quickly). Out went the book definitions and “guided practice” problems and in came an exploration though an inductively-reasoned course with more open-ended problems (fewer of them) that seemed to reward students’ efforts more authentically than the constant stream of “1-23 (odds).”

The second thing that I noticed, though, was that I had less of a script already provided. The textbook takes a lot of the guesswork out of sequencing questions and content. When the textbook goes, all that opens up and it fundamentally changes lesson planning. The lesson becomes more of a performance. There’s an order. There’s info that you keep hidden and reveal only when the class is ready. Indeed, to evoke the imagery of Dan Meyer (@ddmeyer) it should follow a similar model to that of a play or movie.

But, see… here’s the thing about the “performance” of the lesson plan. The students have a role to play as well… and they haven’t read your script… and they outnumber you… and there is a ton of diversity among them. So, when you unleash your lesson plan upon them, you have a somewhat limited ability to control where they are going to take it.

Therein lies the anxiety.

If a student puts together a fantastic technique filled with wonderful logical reasoning that arrives at an incorrect answer, you have to handle that on the fly. It helps to be prepared for it and to anticipate it, but the first time you run a problem at a class, anticipating everything a class of students might do with a problem can be a tall order.

Case-in-point:

The candy pieces made a grid. The rectangles were congruent. Let's start there.

The candy pieces made a grid. The rectangles were congruent. Let’s start there.

The Hershey Bar Problem was unleashed for the first time to a group of students. Our department has agreed to have that problem be a common problem among all three geometry teachers and to have the other two teachers observe the delivery and student responses (I love this model, by the way).

The students did a lot of the things we were expecting. But, we also watched as the students took this in a directions that we never saw coming. The student began using the smaller Hershey rectangles as a unit of measure. One of the perplexing qualities of this problem is that the triangles are not similar or congruent. Well, the rectangles are both. So, as we watched, we weren’t sure what conclusions could be drawn, what questions the students might ask, or how strongly the class might gravitate toward this visually satisfying method.

We didn’t want to stop her. We weren’t sure if we could encourage her to continue. We just had to wait and watch. That causes anxiety. It feels like you aren’t in real control of the lesson.

Adding an auxiliary line changes the look of the problem, "How were we supposed to know to do that!"

Adding an auxiliary line changes the look of the problem, “How were we supposed to know to do that!”

In the end, most of what we anticipated ended up happening. The team trying to estimate rectangle grid areas ended up seeking a different method for lack of precision and everything got to where it was supposed to. The experience is valuable. But the anxiety is real.

This student was trying to make sense of the perimeters and areas

This student was trying to make sense of the perimeters and areas

And I suspect that the anxiety has a lot to do with why the textbooks continue to stay close at hand. When the structure leaves, the curriculum opens up. When the curriculum opens up, the task of planning and instructing becomes more stressful and (for a short time) more time-consuming.

If you are reading this and on the cusp of trying to move away from your textbook, please know that this is the right move. Your book is holding you and your students back. You can do this. I won’t say that there is less stress, but with more authentic lessons, there’s more authentic learning.

Proof and Consequences

A conversation was taking place over at Dan Meyer’s Blog (http://blog.mrmeyer.com/?p=17964) about proofs, which is a topic that I find myself faced with about this time every year.

This isn’t a new conundrum for me. I’ve been working for while now trying to make this idea of proof, which, when compared to the typical form of textbook Algebra I should be an easier sell. But it just isn’t.

Here are some discussions of my previous attempts to sell it. Posts from Nov 2, 2012, Nov 16, 2012, Dec 7. 2012 are a few examples of my thoughts from around a year ago when geometry hit this place last year.

The problem I have is that the academic norms seem to prefer deductive reasoning to inductive and use of the theorem names. Those two things seem important to decide on before starting the journey of proof. If you are going to prefer deductive measurements, it rules out using measurements in proofs and it requires a lot more formal geometric language.

The problem that I see is that to rule out measurements (at least from the very beginning) and to strongly increase the formal geometric language in a way that makes deductive proofs possible from the very introduction of proofs creates… well… what Christopher Danielson is quoted as saying in Meyer’s post… “one of the most lifeless topics in all of mathematics.”

In order to breathe life into the topic, from the experience I’ve had, you need to let students engage in ways that make sense to them at first. The target to start the process is simply to get them comfortable with the idea of designing a functional persuasive argument about a mathematical situation. This requires recognizing that they need to start with a clearly stated claim (preferably something that is provable) and then start supporting it.

I find it helpful to let them pull measurements from pictures first and use those in the proof. The idea of comparing two things by length and NOT measuring them to get the length seems to a lot of kids like we are making the math difficult simply because we want it to be difficult. If they sense there is an easier way to solve a problem, then the explanation for why that method is against the rules had better be very strong, or else buy-in is going to suffer some pretty heavy causalities.

Once they get the hang of making an argument, then we can start by having discussions about what kinds of evidence are more compelling than others. This is usually where the students can figure out for themselves that each piece of information needs its own bit of mathematical support.

Next we can start deliberately exposing the students to different ways of proving similar situations. Triangle congruence seems to be a popular choice. We can have conversations about proving a rigid motion or proving pairs of sides and angles. Eventually certain kinds of explanations become more and more cumbersome. For example, using definition of congruent triangles to prove that two triangles are congruent as shown here:

Do we really need to keep going to find the three pairs of congruent angles?

Do we really need to keep going to find the three pairs of congruent angles?

Then, we can start pushing into shortcut methods. Mostly because those angles are going to be somewhat tricky to find (and why do more work than you need to… the students DEFINITELY identify with that.)

By using this method, I am trying to create what I’ve heard Meyer call “an intellectual need” for additional methods to prove this claim. (Keyword: trying… not sure how successful it is, but I’m trying.)

Then, that transitions fairly smoothly into stuff like this:

2013-10-28 08.28.32

… where we standardize the side lengths of two different triangles and see how many different triangles we can make and in what ways they are different.

Now, the tougher question is whether or not you allow the class consensus following the “Straw Triangle Activity” (which was a gem that came out of Holt Geometry, Chapter 3) to count as proof of the SSS theorem. In an academic sense, now we should “formally prove” SSS theorem. To most of the students, it’s settled. Three sides paired up means the triangles are congruent. What are we risking by avoiding the formal SSS proof? Do we risk giving the impression that straws and string are formal mathematical tools? But wait… aren’t they? What do we risk by doing the formal SSS proof? Do we risk our precious classroom energy by running them through an exercise there isn’t a lot of authentic need for right now?

Am I able to say that this is the definite recipe for breathing life into geometry proofs? Not even close. I am sure there are students who are completely uninspired by this. I can say using anecdotal evidence that engagement seems significantly and satisfyingly higher then when we used to run deductive two-column proofs at students from the very beginning.

But, we’ll have to see what the consequences are as we keep going.

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.”