Spring has Sprung – and so has the French Garden!

So the spring term means two things for my Honors Geometry kids – the technology inquiry project and looking at the French Garden Problem.  So for those of you who are not familiar with both of those I’ll try to quickly fill you in while I talk about how they just happen to so coolly (is that an adverb?  if not I just made it up) overlapped this week.

My Spring Term Technology Inquiry Project is something I came up with three years ago when I really wanted a way to push my honors geometry students into thinking originally while at the same time assessing their knowledge of using technology.  I did a presentation last year at the Anja S. Greer Conference on Math, Science and Technology and the audience loved it.  Basically, I give students an inquiry question (one that I attribute to my good friend Tom Reardon) that they have to work on with technology and then they have to come up with their own inquiry question (which is, of course, the fun part) and explore that with technology and/or any other methods they wish.  I have received some pretty awesome projects in the past two years and I don’t think I am going to be disappointed this year either.

The French Gardener Problem is famously used in my PBL courses at the MST Conference as well.  Everyone who has taken my course knows the fun and interesting conversations we have had about the many ways to solve it and the extensions that have been created by many of my friends – an ongoing conversation exists somewhere in the Blogosphere about the numerous solutions – In fact Tom sent me a link just last fall to a more technological solution at Chris Harrow’s blog. (We’re such geeks).  Great math people like Phillip Mallinson and Ron Lancaster have also been drawn in by the attractive guile of the The French Gardener Problem.  In this problem, the main question is what fraction of the area of the whole square is the octagon that is formed inside (what is the patio for the garden)?

So the other night, after we had worked on this question in class for a couple of days and the students had meet with me in order for me to approve their original inquiry question, a student stops by to discuss his question.  John starts off with, “I can’t think of anything really. What I had wanted to do, someone else already claimed.” (I’m not letting them do a question that someone else has already decided to look into.  So John sits in my study and thinks for a while. I told him that this part of the project was supposed to be the fun part.  I gave him some thoughts about extending some problems that he liked.  He said he had liked the French Garden Problem and thought it was really cool.  So I went back to some of my work and he started playing with GeoGebra.  Before I knew it he starts murmuring to himself, “Cool, cool….Cool! It’s an octagon too!”  I’m thinking to myself, what has he done now?  I go over to his computer and he’s created this diagram:

John's Original Inquiry Question

John’s Original Inquiry Question

I’m asking him, “What did you do? How did you get that?”  He says that he just started playing with the square and doing different things to it and ended up reflecting equilateral triangles into the square instead of connecting the vertices with the midpoints as in the original French Garden Problem.  Then he started seeing how much of the area this octagon was and it ended up that it was……you don’t think I’m going to tell you, do you?

Anyway, it just made my night, to see the difference in John when he came by and the by the time he left.  He was elated – like he had discovered the Pythagorean Theorem or something.  I just love this project and I would encourage anyone else to do the same thing.  Leave a comment if you end up doing it because I love to hear about any improvements I could make.

The Role of Technology in Relational Pedagogy?

So I’ve been thinking a lot lately about technology and learning.  There’s so much in the news about MOOCs, using iPads, schools using technology, etc.  I am even part of a pilot program at my school right now where all of my students have iPads in my honors geometry class and we are trying to communicate at night using Voicethread and the iPads.  My hope was that having a way to share ideas during the evening would lessen the stress of homework problems that students are asked to grapple with in the PBL curriculum would give them more opportunities to throw out problem-solving ideas with each other before class starts so that we would spend less time in class debating different methods of solving the problem (although that’s what I love about class, right?).

But I’m asked as a teacher to find ways to integrate technology into my classroom – but to what end?  I want to find ways to use technology to solve problems, to explore ideas and to help improve students’ understanding of the mathematics.  Not necessarily help them communicate with each other, which is what I’m finding most of the apps out there are for right now – which I am open to – but they are removing a huge part of the learning triangle.  In fact, David Hawkins (1974) wrote about the I-thou-it reciprocal relationships in learning that simply must exist between the learner, the teacher and the subject matter.  He said that if one of the relationships is hindered or dysfunctional in some way, that learning is not optimal.

Hawkins (1974)

So if I interrupt that relational triangle between the students’ communication with the material (and with each other) and with me, using technology instead of discussion and the connection with all three, my fear is that learning is not optimal.  Perhaps the technology could enhance it, but for now I see that it is not truly happening.  My guess is that it has to take time for the students maybe to want for that to happen.

I also just read an article on Edutopia by a guy named Matt Levinson that was entitled “Where MOOCs Miss the Mark: The Student-Teacher Relationship” where it was stated that a lack of mentorship, close guidance or meaningful relationship between teachers and students is what is really lacking in these online courses. Even students who use Khan Academy lectures for “learning” sometimes comment that even though they don’t like sitting and listening to lectures in math class, they would “much [prefer] listening to her math teacher explain the same concepts because she likes this teacher and feels comfortable asking questions and going for extra help outside of class.”

Carol Rodgers (one of my most favorite people on earth) writes about teacher presence and the importance of it in the classroom.  I believe in mathematics class and especially the problem-based mathematics class it is truly essential because in order for students to take a risk with a method, they need to feel supported and safe in order to be open to new ideas and to discuss them with others.  With the open presence of a teacher and mentor, students are not “receiving knowledge” but creating it with others – creating it within those relationships that Hawkins was talking about – maybe with technology or without it.  But  for someone who just spent two years writing about the importance of relational pedagogy in PBL, I find it extremely difficult to assume that without those relationship the same exceptional amount of learning would go on.

Some (hopefully) helpful Mobile Technology (i.e. iPad) information

This past week I spoke with many teachers who are being asked to implement an iPad program in their schools this coming year and feel as though they are lost in the woods. Although their schools are doing what they can to support math teachers in their endeavors, the truth is that the “mobile technology of the future” (i.e. the direction that most technology coordinators say that education is moving) really has not caught up with the needs of mathematics educators. In a presentation I gave last week, I made the distinction between three different types of apps that exist out there for math teachers to use. I believe that the “tool apps” are useful when you want the mobile device to replace an actual tool or a skill that students have learned or that you feel can be replaced but a short cut. Great examples of these are ruler or protractor apps. However, beware of apps that are tools for doing the quadratic formula – just pop in a, b, and c and 30 “practice” problems can be done in 5 minutes (although I could go off on having students do 30 of the same type of problem for homework too). These apps are not necessarily made to aid in the process of learning for students.

Secondly, there are the “review apps” – the ones that are created to help students prepare for standardized tests, name all the theorems in geometry from A to Z, list all the possible types of polygons and their interior angles, etc. These are helpful apps for reference once a student has learned the material and for reviewing for end of year exams, etc. However, once again they do not necessarily aid in the learning of the material.

The third kind of app is what I was truly looking for – these “teaching apps” are really “understanding apps.” They make the process of understanding a concept or whatever is going on in the classroom more productive, efficient, interesting or engaging. I have to say that sadly, these apps are far and few between. I have surfed many a math app blog on the internet and there is no distinction between these three categories and in my mind, teachers want a distinction. Many of the ratings in the iTunes store are made by students so I recommend reading the review and if it says something like “This app is great. It let me do my homework in 5 minutes” (5 stars), my guess is it’s not necessarily the app you are looking for. Here is the list of apps I gave out at my CwiC session that I found useful and within this third categories of math education apps.

The last thing I had to comment on was the fact that the iPad (and mobile devices in general) have a way to go before they catch up with the old Tablet PC when it comes to digital ink. Writing, for a mathematics teacher, is still the easiest way to put equations into lesson plans, tests, board presentations, problems, correcting papers, etc. So although we are not artists looking for the best stylus for drawing or sketching, we actually do drawing and sketching. We are not business-people who take notes during hugely important meeting with clients, but it still is annoying when your stylus makes noise on the glass screen in a meeting with administrators or even in class with students. And having a wrist guard or palm protector that actually works (and doesn’t make the screen move or leave marks) in the note-taking app, is extremely important to us. We also need drawing tools like geometry shapes, coordinate axes, and hopefully (dare I ask) access to symbolic text, like Greek letters.

At the conference I got into a great conversation with some teachers about different styli and which were the best. I found a Great Stylus Video Review online that I highly recommend if you are looking for a new stylus for your mobile device for writing or doing mathematics. I think I might actually order the Maglus from overseas to see how good it is.

Keep in touch about the mobile apps you are using, cause I’d love to hear about them. I do believe that someday the devices and their technology will catch up with the needs of math teachers, but for now, I sort of miss my Tablet PC and OneNote for writing – :( But I am loving playing with all these new apps. I’m like a kid in a candy store, but hopefully more productive.