Tag: inquiry

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.

Why Can’t We All Just Get Along?: Some Inquiry Math Classes are not Content-less

Maybe it’s just how I am, or maybe I’m just always worried about what people are going to say about me, but I am hesitant to criticize other teachers publicly in the blogosphere. I’ve always felt this camaraderie with others once I’ve learned they were a teacher even if we are very different from each other – different disciplines, different pedagogical styles, different countries – there are still fundamental commonalities that even public and private school teachers have.

I just finished reading a KQED blogpost entitled “Do You Have the Personality to be an Inquiry-Based Teacher?” that sort of summarizes the theoretical qualities that the author feels a teacher who would teach with IBL would need to exhibit in order to successfully run a classroom. It’s kind of interesting – I’m not sure I agree with it, but respect the author for putting his ideas out there. I’ve been an inquiry/problem-based teacher for almost 20 years and I don’t think I exhibit all of the qualities listed, so I’m not sure it’s quite true.

Anyway, that’s not the point – at the end of the blogpost there are about 11 comments from people who are educators and many of them are quite negative and even degrading to the author:

“I earned a Ph.D. in Educational Psychology, but phrases like this one still baffle me: “…the divide between a transmission model and an inquiry model…” ”

“First, we need to make sure that we have at least a rudimentary understanding of the language in which we will be teaching. Second, we need to make sure we can write.”

“That is what’s wrong with you teachers.You want to do it your way.”

“Some of us have been doing this for decades, where were you?”

Whoa, Whoa, Whoa…cowboys…hold your horses. This guy is just writing an essay about something he believes in. What kind of role model are we being for our students if this is how we are reacting to something we don’t agree with? What happened to civil communication? I totally agree that people are allowed to comment and voice their opinions on someone else’s opinion, but there has to be a way to do it with respect and decency.

So I am going to try to model what I would like to see as a response to something I actually do disagree with. Here is a blogpost by a very respectable Professor in Canada, who I have to be totally honest, I do not know at all. I tried to learn as much as possible about him before writing a response to his blogpost in order not to make any assumptions about him (and not make a fool of myself in doing so), so I may be wrong about some of this information because I garnered it from different websites. It seems he is a research mathematician who is currently studying to get a teaching degree, but who lectures for mathematics courses at the college level. I cannot ascertain if he has any experience teaching at lower levels (like elementary or secondary). From his blogpost it does seem like he takes pride in the amount of background research he does, which again is very respectable and I appreciate in bloggers. He seems to care a lot about student learning and from his opinions on his blog he seems to lean towards being a behaviorist and cognitive theorist in terms of learning theories.

His latest blogpost is titled “The Content-less Curriculum” and it is a critique of the movement towards 21st Century skills being a part of the mathematics classroom. It does sound like Prof. Penfound is implying that with the inclusion of “soft skills” of collaboration, critiquing others work, problem solving,communication, etc. (i.e. the MPS for the CCSS) there must be a loss of mathematical content. In fact, he says that

“there must be a trade-off for the inclusion of “soft skills” activities into an already packed curriculum. So what gets removed from the curriculum then? Content knowledge.”

I would respectfully, but wholeheartedly disagree with this. By teaching with the PBL curriculum that I use, I have all of the college prep geometry curriculum I desire and I also concurrently am assessing and teaching the skills of problem solving and the so-called “soft skills” that he is implying are an add-on. I still give quick quizzes to make sure that students are up on their basic skills that are so important for basic problem solving (or else they wouldn’t be able to do the open-ended problems they are given).  The mathematics that students leave my courses having experienced is rich and leaves an impression on the way they think.

Making blanket statements about teachers implying that we all make choices that are not based in research or good practice is just not true. I actually invite you Prof. Penfound to come visit my classes and see my IBL/PBL classes in practice and let me know what you think of your opinions of the rigor of the mathematics that is discussed. Although we are most likely at different ends of the spectrum in terms of learning theories, I do believe that students have different needs and try to work with kids’ learning needs individually. However, I do believe as @danieldmccabe does that there are going to be new outcomes required of our ever-growing diverse body of graduates in the near future (or even present). I also have to say that I have thought rather thoroughly about the implementation of a teaching program which includes “soft skills” and even wrote a dissertation on it.

It is possible to balance content and practice skills and it is what I and many other classroom practitioners strive for. I do not deny that there are some practitioners out there that are confused about what problem-based and project-based learning outcomes should be especially with regard to secondary mathematics, but that is a subject for another blogpost.  The balance between content and practice skills we should strive for does not mean that one is more important or less important and in fact they both need to be assessed with the ultimate goal being to create independent problem solvers. From my experience this does not necessarily happen in a classroom where the educator does not take into consideration the so-called “soft skills.” But that statement is, of course, based on my 25 years of anecdotal classroom experience.