For the Intro to Geometry unit, we go through the definitions, symbols, and nomenclature for how to represent objects in Geometry.  They all know what it means “to cut an angle in half” from middle school, but our concept of an angle bisector is more detailed and nuanced.  I used to do this lesson with pencil and paper in a very traditional way, but to change it up we did the entire set of notes in Geometer’s Sketchpad (not Geogebra because I haven’t had time to learn efficient use of it).  A page of notes looked like the following:

Why?

As I can see it, there are two benefits of doing the notes this way:

1. The dynamic nature of sketchpad is nice for the students building an intuition of how geometry works.  Does the angle bisector always stay inside the angle? Do midpoints move if the segment changes? Etc.  Obvious answers to math teachers, but these concepts are not hard-wired into the average 10th grader.
2. It’s an excellent intro to how to use sketchpad.  They practice just about all the skills necessary to quickly research a topic with this software.  I could now have them create a triangle and see if the centroid is always inside the triangle.  I could ask them to find the sum of all the exterior angles in a pentagon.  I hope that they have gained a knowledge of how to construct and measure with sketchpad.

Disadvantages:

1. It was a battle between going fast enough to keep the quickest students engaged, and not losing the slower process students.  I ended up going at a good pace, and telling the students to just keep up with me, even if you can’t type everything in yourself.  Some of the students still can’t touch type (!), so that was a disability for them.  At the end I passed out a printed packet of the finished product, so they had a full set of notes.  This also stunk because I think it would be cool for them to just print theirs out, but that would have required 3x the amount of paper and 100x the amount of hassle.

Overall I am still battling myself to find a proper mix of  and old-school thought of “Get the Math done!” to the new-school techie thought of “Gee-Whiz that’s cool”.

Have any of you tried such a technique?  Where have you done this differently?

p.s. As usual, in education, the problem is tremendously difficult, there is no easy way of measurement, and no clear solution.  I like this job.

I have access to Geometer’s Sketchpad at school; its installed on a bunch of laptops and in a couple of computer labs.  Geogebra is a similar program but I do not have much experience with it.

For the past 4 years I’ve done various “labs” with the Geometry students where they create a set of objects in sketchpad and then measure the different characteristics.  From these measurements they are supposed to write a corresponding theorem to fit the data.  For instance they would create the following object: They would then measure the angle CAO and the arc ADB to find out that the measure of the angle is exactly half that of the arc.  The top quarter of students would have this written down after putting some words in their pencil: “The measure of the angle formed by a tangent and a chord is half the measure of the arc that is inside“.  My “putting words in their pencil” is the following leading question:

(the full document and its 49 steps are here: http://drop.io/circleAnglesLab )

49 steps? Are you kidding?

Am I giving the students too much help?

(or as Dan Meyer would say: “be less helpful”)

The reason there are 49 steps to the document is that this is the first time the students are seeing geometer’s sketchpad and it’s less than intuitive on how to measure an angle or an arc with this software.  The document was created when I thought that it’d be best to give the kids as precise of a path as possible to the truth.

Overall I think this type of project is a good idea for how to teach some sections of geometry, but I’d like to be less focused on the steps and more focused on the material.  Many students would carefully go through the steps and get to the end with the theorems written correctly, but still mess up the proper theorems on the assessments.

Where should I go from here?

Maybe I should have an intro “lesson” on how to use sketchpad to measure circles (demo how to measure an arc) and then set them free with a goal (Find the measure of an inscribed angle in relationship to an arc).  High expectations are an excellent thing in a classroom, but am I asking the all the Oilers to be The Great One?

Last Question.

For those Sketchpad and Geogebra experts out there, would you recommend Geogebra over Sketchpad?  I haven’t had enough time yet this summer to start playing with Geogebra and so I only have a very basic understanding of it.  Thanks!

Mostly this summer I will be spending time with my family. On Mondays and Fridays, I’ll be home with my two daughters. On Wednesdays I’ll have them plus my 16-month old son, plus my wife will have that day off. On Tuesdays it’ll be just the boy and me. So I plan lots of trips to the zoo, the various parks around here, and so on.

I still have plenty of time to work, and I have a few projects for the summer.

First, I need to get ready for my Geometry class this fall. I am making the move from Geometer’s Sketchpad to Geogebra this fall, and although I took a minicourse at the ICTCM on Geogebra, I still need to work on my skills before I teach with it. Also, I need to figure out exactly what I am going to teach. I’m going to be using Euclid’s Elements as the textbook for the course, eschewing commercial textbooks for both monetary and educational reasons. But I’m not totally sure what I’m going to have students do, exactly. So I’ll be reading through the Elements and possibly thinking out loud here on the blog about how to incorporate a 2000-year old mathematical work with modern open-source dynamic geometry software in an engaged classroom. I’m calling it “ancient-future geometry”, whatever it turns out being.

Second, I’ll be working on our dual-degree Engineering program to try and make it a little easier to schedule and complete. This is hard-core administrative stuff, interesting to nobody but a select few geeks like me.

Third, I’ll be working to further my programming skills with MATLAB and Python. I picked up a lot of MATLAB programming to get ready for the course this past semester, but that seemed only to highlight how much more I needed to learn. And I watched enough of this MIT computing course over Christmas break that I want to do the whole thing now that I have some time.

Fourth, I’ll be attending the American Society for Engineering Education conference in Louisville next month. Part of that experience is a day-long minicourse titled “Getting Started in Engineering Education Research”. I’ll be taking my participation in that minicourse as the kickoff to a concerted effort to get into the scholarship of teaching and learning. Along with the minicourse I’ll be reading through some seminal SoTL articles this summer, and probably blogging what I’m thinking.

Fifth, and finally, I’ll be mapping out some incursions of the inverted classroom model in my Calculus course this fall. More on that later as well.

For now, my family and I are heading out to Tennessee on vacation to visit family and hang out. I’ll be off the grid for a week or so. Enjoy yourselves and stay tuned!

Image via Wikipedia

[Ed. note: This post was originally written on March 13 while at the ICTCM, but I ran out of time on my $12.95 per day internet access before being able to post it and only now have had the chance to get back online. So it's about 36 hours out of sync.]

Slower day at the ICTCM than yesterday. For one thing, I took some time out in the morning to get the MATLAB course prepped for Monday; and I needed time to finish some grading in the afternoon. But I manage to have a pretty productive day nonetheless.

The main event — one of the primary reasons I came here — was a Geogebra 3.2 minicourse this morning. I’ve been a diehard Geometers Sketchpad user for a long time, but after becoming aware of Geogebra lately, I began to wonder if it might be time for a switch. I have no problem with the usability and features of Sketchpad, but if there’s free software out there that is pretty close to the same quality, the possibility of simply installing it everywhere (like we’ve done on campus with Winplot) is pretty enticing. The question was whether Geogebra’s features and usability matches up well with Sketchpad’s.

After the minicourse, I’d say the answer to that question is definitely “yes”. Particularly impressive is Geogebra’s ability to export entire constructions to HTML as an interactive web page. I have some definitely plans for this kind of thing for the class now. There’s more to learn — unfortunately we didn’t go very deep with the software in the minicourse — but definitely Geogebra will be the software platform for the geometry course this fall. Now I have to decide on a textbook — or to go without. Hope to blog on that later.

Also today I attended a session on using clickers in mathematics courses. I’ve been following Derek Bruff on Twitter for some time (he’s an assistant director at the Vanderbilt Center for Teaching, where I used to be a Master Teaching Fellow) and have gotten interested in using clickers through his work with them. This was a general survey talk, but very well done and it definitely increased my interested in folding clickers into my teaching mix at some point.

Overall the ICTCM is one of the better conferences out there for people who are interested in math, education, technology and the intersections between these. Look for the announcement for ICTCM 2011 coming soon!

• I was a psychology major at the beginning of my undergraduate years and made it into the senior-level experiment design course as a sophomore. In that course I learned how to use SPSS (on an Apple IIe!). That was an “extra” that I really enjoyed, perhaps moreso than the experiment I designed. (I wish I still knew how to use it.)
• In my graduate school differential geometry class (I think that was in 1995), we used Mathematica to plot torus knots and study their curvature and torsion. Learning Mathematica and how to use it for mathematical investigations were the “something extra” that I took from the course. Sadly, the extras have outlived my knowledge of differential geometry. (Sorry, Dr. Ratcliffe.)
• In the second semester of my graduate school intro abstract algebra class, my prof gave us an assignment to write a computer program to calculate information about certain kinds of rings. This was a small assignment in a class full of big ideas, but I had to go back and re-learn my Pascal in order to write the program, and the idea of writing computer programs to do algebra was a great “extra” that again has stuck with me.

Today I really like to build in an “extra”, usually having something to do with technology, into every course I teach. In calculus, my students learn Winplot, Excel, and Wolfram|Alpha as part of the course. In linear algebra this year I am introducing just enough MATLAB to be dangerous. I use Geometers Sketchpad in my upper-level geometry class, and one former student became so enamored with the software that he started using it for everything, and is now considered the go-to technology person in the school where he teaches. In an independent study I am doing with one of my students on finite fields, I’m having him learn SAGE and do some programming with it. These “extras” often provide an element of fun and applicability to the material, which might be considered dry or monotonous if it’s the only thing you do in the class.

What kinds of “extras” were standouts for you in your coursework? If you’re a teacher, what kinds of “extras” are you using, or would you like to use, in your classes?

The lesson plan can be found at http://docs.google.com/Doc?docid=0AXxJ1gaN0eqJZGYzd2M1bV8zNGhwZno3emM0&hl=en

I started the lesson by first demonstrating the construction on the board with a large compass, then on the document camera with a regular compass, then having the students try it on their own on a handout with 3 practices, while I walked around and helped students.

Reflection

Constructions:  The state expectations are that students make constructions with software or with compasses.  I can’t imagine any workplace wanting my student make parallel lines with a compass, but I can think of many positions where they have to do it with software.  While I’d taught the students the constructions on paper because of the portfolio piece, I will promote with the other teachers that future assessments of construction have a software option.

In chapter 5, we will cover constructions in detail.  Thinking over how this lesson went, I will block the computer lab for those days and plan on giving the students ample time to get comfortable with Geometers Sketchpad.  If there are students without AUP’s (computer use agreements), I’ll have to have them bring compasses and construct on paper.  One thing for sure, I’ll plan extra time for developing comfort with either tool.

ScienceDaily (Jan. 5, 2010) — A new study co-written by a University of Illinois expert in math education suggests that incorporating technology in high school-level geometry classes not only makes the teaching of concepts such as congruency easier, it also empowers students to discover other geometric relationships they wouldn’t ordinarily uncover when more traditional methods of instruction were used.

I’ve used Geometer’s Sketchpad for about 6 years, and I really learned a lot by tinkering with different shapes, doing “shake” test, etc.  Highly recommended.

First, head over to

http://www.chartwellyorke.com/sketchpad/gspdemos.html

and download the “demo”. If you read, the demo is literally the full version without the ability to save or export to web documents. However, don’t worry, it doesn’t end there.

Basically, what I did from here is download several versions of a crack for this program. The one that worked for me was 4.06 I believe. Basically, I switched the .exe in the demo with the crack, and whadyaknow? Works flawlessly with all “restricted in demo” features gone. It works great in both Linux and Windows, with a slight lag in Wine, but it is barely noticable.

However, I do know that these files might be a bit difficult to find, so I’ve taken the liberty of providing it for you, this is the full version, and will work. As far as I know, its one of the only working pre-cracked copies online.

Edit: I got this working with Version 4.1 . Your AntiVirus might complain about a trojan, ignore it,  it’s a legitimate crack.

http://www.box.net/shared/vo1d0jyi9n

Enjoy!

As far as I can tell the only difference between Alg II and Pre-Calc is that trig is taught during Pre-Calc and Pre-Calc introduces the concept of the limit. Functions are developed a bit more rigorously too.

The first semester of Algebra II is mostly a repeat of Algebra I as they’ve forgotten it with the year “off” during Geometry.

Why not then teach Geometry first? I’m talking about plane and solid geometry with an emphasis on reasoning, and right angle trig. Obviously there would need to be some supplementing needed (work with radicals, solving equations). Most students have “seen” the solving of equations in 8th grade (Have they mastered it? No, of course not).

I completely agree. It seems to me that the reason Geometry gets sandwiched between Algebra I and Algebra II is that people want to use algebra concepts in geometry. But I think that doesn’t necessarily have to be the case. If you look at the source — Euclid’s Elements — you will not find a drop of algebra in it. All the concepts that we, today, would label as being algebra or number theory or what-have-you are just latter-day retrofittings of Euclid’s ideas. Euclid himself phrased everything in terms of geometry, with the algebra and number theory done in terms of commensurable lengths and other geometric terminology. I wouldn’t go so far as to say Euclid knew nothing of algebra or number theory, but if you follow Euclid you don’t need algebra, as we know it, at all in your geometry.

That would leave a geometry course that is mainly about logical reasoning, cogent organization of facts, objective deductions from data, and clear exposition of an argument. One might add to this list the art/craft of forming conjectures from experimentation and then writing an argument in favor of your conjectures, which is astoundingly simple these days thanks to Geometers Sketchpad and other fun, low-cost dynamic geometry software packages. (My students who use Sketchpad in their student teaching report, to a person, that students really turn on when they use Sketchpad and do some very good mathematics, for 8th-9th graders.) This sounds like precisely the kind of foundation, and buffer zone, that students need to acquire before tackling algebra with a view towards understanding how it works rather than just memorizing facts. (Indeed, memorizing facts in algebra is quite hard unless you understand why the facts work.)

Of course, if you ask ten people whether they liked their geometry class in school, eight will probably say “no” and seven of those eight will say it was because of “proofs”. But I wonder what that really means. Perhaps, having gotten a taste of equation solving in algebra and therefore acquiring the “there’s only one right answer and I have 30 seconds to find it” mentality about mathematics, they are spoiled for ever encountering mathematics as it really is (which is something that geometry is a lot closer to than algebra I). Perhaps they had a geometry teacher who was not really good at, trained in, or interested in math at all — or someone who was like so many teachers out there who “just love kids” but who choose not to translate that love into teaching their kids how to think well.

But I think if you put a geometry class like what I described above into the hands of a competent, mathematically astute teacher with a mind to help his/her students become excellent thinkers, a year of that could very well change a generation of kids.

The full instructions on how to do so are here (search for “Tables”). It’s relatively painless. Select the measurements, go to the Graph>Tabulate menu. There you go! To add more measurements, double click on the table. Done.

I really like this as a way for students to organize and analyze the effects of changes to the figure. I think too often they just click and drag on things without stopping to think about what is really happening. Of course a good worksheet to supplement the activity is necessary. I’m still working on that part.

If you’d like this file, you can download it here. You will need Sketchpad though. Just for fun I tried the “Save As>HTML JavaSketchpad” option. The full features aren’t available (like the table) but here it is (Java compatible browser required). I need to play with this option some more. It is on my list of things to figure out.

Speaking of which, does anyone know how to embed these Java Sketchpad files directly into a WordPress post? My trust aide gave me some code to do so, but WordPress didn’t like it.

Today will be no less busy:

• 8:00-8:45: Session on handheld calculating devices over the last 30 years and how they have changed teaching. Very interested in this talk; I’ll have more to say about some of the handheld technology I’m seeing here.
• 9:00-9:45: Session on using Maple 11 in the advanced calculus and modern algebra classroom.
• 9:45–10:30: Exhibit hall surfing.
• 11:30-12:05: Session on labs in mathematics classes.
• 12:30-1:15: Session on using Geometers Sketchpad alongside computer algebra systems.
• 1:30-2:15: Session on Winplot.
• 2:30-3:15: Take a break!
• 3:30-4:15: Session on blogging with concept maps. Two of my favorite things put together, so this ought to be fun.
• 4:30-5:15: Haven’t made up my mind yet — either a session on CaluMath or a session on using Geometers Sketchpad in calculus courses.

Unfortunately the internet access I am paying $10 a day for isn’t wireless — or at least, there is wireless but yesterday it didn’t play nice with me. So I won’t be blogging continuously. Which is probably a good thing because I need to pay attention at these sessions. Speaking of which, it’s time to head down to the first one.

I’ve been bicycling since I learned to balance a bike. Not only great for recreation and competition, cycling is the world’s most efficient form of transportation. Bicycling is based on a simple and elegant technology that’s the most effective and environmentally sound method of travel for short trips.

My enthusiasm for bicycling is rivaled by my enthusiasm for The Geometer’s Sketchpad, another easily mastered and elegant technology.

Apparently he’s going to be posting Sketchpad projects along the way as part of the blog. Unfortunately the route doesn’t come through Indiana, but if you’re on his way then you can have him come in and meet with you.

By the way, expect more blogging about Sketchpad in the future from me too, as this is what I am using instead of a textbook in my upper-level geometry course this fall.