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.

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:

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.”

“Truly wonderful, the mind of a child is”

photo credit: Flickr user “sw77” – used under Creative Commons

I agree with Yoda. (The title of this piece is a quote of his.)

This morning, as she was finishing her cereal, my daughter was completely taken by the reflection of her cup on our table. She was passing her hand under the table and above the table and, as is her custom, talking the whole time.

Then she fired out a wonderful question.

“Daddy, why is the reflection upside down?”

My daughter asked a fantastic question. Would the average geometry student be able to answer it?

In my head, I thought that my students should definitely be able to hammer out a pretty reasonable response to that question by the end of the first unit of geometry.

And thanks to my daughter, they will.

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.

Motivating Learners

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.

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.

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.

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

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.

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.

Double Stuf Oreos: Are they really double the Stuf?

Double Stuf: A factual statement or clever marketing trick?

I was inspired by this post by Nathan Kraft (@nathankraft1) in which he engages his staff in a question about Oreo cookies. (Mr. Kraft was quick to inform me that Christopher Danielson (@Trianglemancsd) was the inspiration for his post. I do want to give credit where it’s due.)

I decided to see what my third hour students would do with it. So, last week Friday I showed them the picture and we started discussing the a variety of aspects of Oreos (some of which were more useful than others). Then, it happened. One students asked:

“Is the stuff of a Double Stuf really double of the stuff in a single stuff?”

The beauty of this activity is that the students were able to become involved in the formation of the solution process. They practically all had a prediction. First idea, would double the stuff be twice as tall?

It didn’t appear to be double by height.

The above image represents what multiple students observed. It was an awesome opportunity to discuss conclusions. What conclusion can we draw from the observation we just made?

Either it was double the stuff and it wasn’t manifesting itself in its height, or it wasn’t double the stuf. (Often, the student’s original predictions colored their conclusion to these observation.)

Double by what measure? Mass?

Next idea was mass. Gave in impromptu call to Mr. Corcoran, the chemistry teacher, who loaned us some scales. But what do we measure? The whole cookie? That opened up another important question? Is the same wafer used for both the standard and the double-stuff?

After some quick diameter and mass measurements, it seemed like there was no meaningful difference between the two. But, just to be safe, each student scraped the Stuf from a standard and a Double Stuf and set to the scale to get a mass measurement.

What math class looked like today.

Then we compiled the results.

The mass of the Stuf scraped off a sample of standard and Double Stuf Oreos.

Each group took a moment to deliberate and concluded that, for the most part, it seems that the Double Stuf is appropriately named. Some groups seemed to think that, if anything, the Double Stuf contained more than double the Stuf.

This activity contained so much of what makes contextual, collaborative learning valuable. Authenticity, source of error, conclusions that were not clear, but needed to be discussed. Students needed to listen, speak and rephrase when others didn’t understand.

It also had the beautiful feature of me not knowing the answer and they knew it. So, there wasn’t the temptation to treat me like the math authority, as though all math learning begins and ends with the Teacher’s Edition.

And for an added bonus, the AP Stats class meets next door at the same time and so, we were able to strike a deal to rerun the trial with the guidance of the stats class for a broader, students-teaching-students experience.

I’ll report back with our findings.