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.

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Our Geometry Support Site

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So, I’ve talked about Khan Academy before. They make instructional videos for students to watch and learn from.

 

I’ve talked about #flipclass before. In this model, the teacher makes videos for the students to watch and learn from.

 

I like the idea of students having instructional videos to watch and learn from. But, in both Khan Academy (and other related sites) as well as flipped classes, the students are recipients of the videos. We decided that we didn’t want them to simply be recipients. We decided that we wanted them to be producers of the resources, too. So, we decided to just let the students create the math help videos.

Here’s what they did: Today I unveil the @PennGeometry Beta for your review and feedback. Currently, there are videos created from one practice test about triangle similarity. We would like to expand and continually improve.

 

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All of the videos are designed, produced, and completed by our students. Some of my students were sensitive to the boring math video and tried to add their personality to the video. We would love some comments, “likes”, and constructive feedback. Please check it out. Eventually, we want to add our voices to the overall mathematics community to support those in need.

Check it out and let us know what you think. We look forward to hearing from you.

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.

Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem

I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I’ll leave you to read it if you are curious what the problem is.)

From my perspective, what makes this problem interesting for the students is the ease with which it is communicated and the complexity with which is it solved. It seems quite easy. The answer is fairly predictable, but the students quickly found out that if they were going to solve this problem accurately, they were going to need two things:

1. A way to organize their thoughts and,

2. a way to verify their answer.

As long as the solution process included those two things, the students ended up fairly successful in the process of this problem.

I submit to you four examples of student solution structures. They all look different, but they all have one thing in common: The students could tell you with absolute certainty what was on the page and that they were right. (I withheld judgment from the correctness, but I will say that their final answers were all pretty much the same.)

This one includes a bit of a floor plan

This one includes a bit of a floor plan

In this first one, shown above, this pair of students decided to draw a floor plan with each of the pencil sharpeners (and the bucket they were tossing their leftover pencil nubs). The solution process progressed from there.

Guessing and Checking

Guessing and Checking

These two students are classic minimalists. They worked to guess-and-check, which requires a bit more skill than it might seem. They needed to decided what number they were going to guess (sounds like an independent variable) and how to check their answer (sounds like a function producing a dependent variable). They weren’t using the vocabulary. It didn’t seem to be a problem for them.

Guessing and Checking

Guessing and Checking

These two were a little more formal with their guess-and-check process, and they were using the terms independent and dependent variable. For the record, the time was independent and the number of pencils sharpened was the dependent variable.

A graph of multiple equations (where the solution does not include finding an intersection point)

A graph of multiple equations (where the solution does not include finding a single intersection point)

These two struggled a bit to convince themselves that they were on the right track as they completed this graph. They were discussing the need for the graph to be accurate (essential), how to interpret the three sets of points (one set to represent each pencil sharpener). Eventually, the were able to find the spot on the graph when the numbers added up properly. The number along the x-axis represented the final time. (Once again, they didn’t call time the independent variable, but it looks so nice along the x-axis, doesn’t it?)

Each of these students would have likely looked at the others with confusion, yet their answers largely agreed. What it required for them to solve it wasn’t me, as teacher, telling them how. On the contrary, it required me, as facilitator, designing a problem they would engage with and then giving them the resources to make sense of the problem with each other.

CD’s on a Desk

A few weeks ago, I asked the students to figure out how many CD’s it would take to cover their four-desk pod. I didn’t tell them anything more than that.

How would you solve that problem?

I was very impressed with what I saw.

1. The questions: Do we need to cover the whole pod? Can they overlap? Can they hang off the edge? What if there are little bits of desk showing through? (Excellent questions because they all change “the answer”.)

2. The methods: All began by measuring their desks. Some chose to leave length and width separate, some chose to multiply into area. Some researched the dimensions of the CD and had to decide how they were going to use it.

3. The answers: There were several recurring ranges of answers, but in terms of the numbers, there were probably 15 different final answers and the students were okay with that considering the assumptions were different.

All of that reflects a TON of great math thinking and problem-solving, but the most convincing solution to many of the students was arguably the simplest. (Isn’t it always that way?)

Here is a photo of the solution:

Photo Credit: Heather Roadcap – Used with permission

The solution: Get a CD, trace it carefully, and count. Can you argue with that?

The Art of Mathematics: Kolams

Kolams are a type of local art from South India. Read about them at the Wikipedia site for Kolams (it is a quick read, but a very interesting one).

What interests me about this kind of art is the symmetry. Below is a slide show demonstrating more Kolams. What kind of symmetry to they have? Are there examples that you can find that have more than one kind of symmetry? (Don’t be afraid to do some hunting around on the web to find some goodies.)

Let’s have a conversation in the comments about the different types of symmetry that exists in different Kolams.

Here is a slide show of different Kolams from Slide.com user Sravani. (If you like the slideshow, drop her a comment and a 5-star rating.)

Oh, and yes… it goes completely without saying that if you make one of these (especially if you use the rice powder or colored sand), I want to see a serious photo of it. Link your Facebook, put it on Flickr, no me importa, just get it to me.

Proof by Crayon: An Inspired Problem, Part I

I have WordPress Blogger (and Ph.D. mathematics student) Jeremy Kun to thank for inspiring this post. His original post “Graph Coloring, or Proof by Crayon” not only lent me the title, but a ton of fantastic insight into the many, MANY different applications of this kind of mathematics. So, soon-to-be Dr. Kun, many thanks and I hope that you enjoy my take on this.

All right, the first part of this problem is a challenge. Print out (or transfer into paint or some other photo editing software) the map shown below. The challenge is to color the map in as few colors as possible making sure that no two states that touch each other are colored the same color.

Now, do the same thing with the Michigan county map shown below.

Now, do the same thing with the European map shown below.

Now, when you have done that, I want you to put into the comments the number of colors you had to use to meet the requirements of the problems for each map.

Part II will come when we get some data to work with.