Posts Tagged 'Geometry'

Beginning proofs as groupwork

Today in Geometry students were taking their first crack at making up proofs. I have modelled this for them, walked them over the pons asinorum and tried to show them how we can use a deductive argument to see how our compass-straightedge constructions work. They have done a bunch of the classic SSS etc. problems and should know the triangle congruence theorems. They were ready, but I knew that it would be hard for them, so I wanted to have them work in groups.

I break my classes into groups nearly every class, if only for a short investigation. But I’ve been feeling unsatisfied, and recently returned to the book Designing Groupwork (Cohen) for some deeper insight. One thing the author stresses is the importance of assigning individual jobs to help break down students’ prejudged status structures. In an effort to bring the groupwork up a notch, and to be really clear and formal about what makes a mathematical proof “right,” I created both a scoring rubric for student proofs and a corresponding set of group member roles. I made up a list of simple proofs I thought the students could handle. The whole thing is a one-page (front-back) sheet, found here.

How it worked: I pitched the task to kids as their first real attempt at mathematical proof. They were actually sort of excited, and nervous. I told them their work would be scored and entered in the gradebook. ( I don’t usually do this, striving for a more righteous SGB, but sometimes it helps to raise the stakes just a tiny bit.) I provided one last model so they could see the level of clarity and justification I expected. Then I shuffled the cards and made groups of four. Each group got a sheet with one of the proof tasks highlighted. I put their names by the roles. And they got after it with the whiteboards.

All the groups were able to write a correct and clear proof, working together. One group in one section took a lot longer than the others, so kids had to do some other problems while they waited. The presentations were pretty good, though lacking in the class feedback department (which I encouraged, but didn’t require or formalize).

The whole lesson, with launch, group work, and sharing, fit into an 80-minute block just right. There are a lot of things left to prove on the list, and I plan to do the lesson again after Thanksgiving. I anticipate that they will be much faster, and more confident. Then, eventually, I’ll be asking them to come up with proofs without the group support, and I plan to use the same rubric (without the participation component) to assess them.


Compass-edge constructions, curiosity

I’m introducing the basic compass-straightedge constructions in Geometry this week. For the most part lessons seem OK; it’s helpful that everyone has something to do. Busy hands are less the devil’s playground than bored hands. My biggest disappointment is that only one or two kids in each section seem to wonder – like, at all wonder – why something like the angle bisector construction works.

This is a big contrast from last year, when I taught at an alternative high school. There, the students were historically low achievers in math (and most other subjects) and in many cases had done some construction or other labor. Hands-on geometry was the best thing I did with those kids. For them, finding the balance point of a cardboard triangle was, like, very cool math. And what I loved was they always wanted to know why something like that worked. It was a very natural, in-the-world kind of curiosity, that felt to me like a true mathematical instinct.

My geometry classes this year are mostly composed of what we would classify as better students. They will undoubtedly be more successful on average on the state end-of-course math test than the cohort I had last year. But I’m waiting – impatiently – for their curiosity to emerge. And I’m wondering what things I can do to inspire and nurture that quality in them.

A joke for hard times

So, in Geometry we hit the ground running with a Geogebra lab which I was pretty excited about. The kids have to make some different polygons, and I get to try to “break” them with the move tool. So, an isosceles triangle should stay isosceles even though we stretch and drag it around. This requires building it in one of several different ways (using a line of symmetry, or fixing two points to a circle to preserve a common distance, etc) which kids have to be creative to find. If you haven’t played around with Geogebra yet, this is a great way to get going with it. One of my goals is to have my students proficient and comfortable with the program, starting in the geometry year, and this lesson was supposed to serve that goal whle helping kids get prepared for the upcoming test on vocabulary, polygon classification, etc.

I teach two sections of Geometry and they meet on alternating days. Monday, the lab went pretty darn great. I did a launch using the classroom projector, and we headed out. In the computer lab, kids were engaged, there was a good mix of sharing and independent work, several kids had breakthroughs, and the lesson differentiated itself fairly well. I was happy. One item on the list is a triangle with a segment from each vertex to the midpoint of the opposite side. We had sketched this during the bellwork and wondered whether those segments always had to meet in a single point. Opinions differed. Now they were seeing that the medians always did meet, no matter how they morphed the triangle. I intended this as a setup for introducing inductive and deductive reasoning, and it seemed to be working. We had a a little discussion about how you can really know something. Joy!

Tuesday was a whole different story. The class is a little larger and has a lot more attitude. Still, I felt ready, emboldened by my success on the day before. Well. First off, I got a glitchy launch of GGB in the classroom, it kept freezing, and I had to relaunch twice. Kids were like, “um, this is lame.” I soldiered on. We got through it and headed to the lab.

Things didn’t exactly soar. Several kids were way not into it, and let everyone know. The toughest thing about this group is the intelligent, high status kids with shitty attitudes. One of them, H., was clearly feeling way too cool for this. She has the potential to be a great math student but has already let me know all she cares about is the grade. Finally I got them going and most kids were making progress.

We get pretty far along and I notice several kids have made the triangle with medians. I get a discussion going about how they always seem to meet in one point, no matter how the triangle changes. “So we can see how that always seems true, but is that the same as knowing that it’s true? As we go forward we will gain the tools to prove that it’s true, which is the same as understanding why it is true.” Pretty nice lead-in to reasoning and proof, right? Lockhart-esque, even? Wrong. Says A-student H.: “Things just are. I don’t really care why.” General agreement from her peers in 5th block Geometry.

Which brings me to my joke.

Jesus was out walking when he came upon a man crying. “What is wrong, my son?” “Once I was a great violinist, Lord. But last week I crushed my hand, and now I am unable to play.” And Jesus laid his hand upon the man, and Lo, he was healed and able to make music once again.

Later Jesus came upon a woman who sat crying. “What is wrong, my child?” “Once I was a painter, Lord, but now I am blind and all art is lost to me.” And Jesus laid his hand upon the woman, and Lo, she was healed and went on to paint once again.

Still later Jesus came upon yet another man who was crying. “What is the matter, my son?” “Lord, I am a high school math teacher.” And Lo, Jesus sat down and wept with him.

Wrestling with homework (Algebra 1, Geometry)

I started this year with a homework policy that came out of tempered idealism. I believe that in high school kids should gradually take ownership of their learning, and that includes practice. I think they are capable of doing a small amount of practice for no reason other than it will help them learn and retain new skills. No points! No homework grading!

On the other hand, I knew that a certain percentage of kids would work the system however they could, including to their own detriment. No homework grading? Cool! No homework!

Then there are some kids who probably will not do the homework whether you grade it or not. Some of them have chaotic home lives. Homework inherently discriminates against these kids. Many of these kids are the ones we will work ourselves to the bone to prevent from failing.

So, what to do? I feel I need to encourage practice, and responsibility. I do not want to grade 135 kids’ homework. I don’t want to “check in” homework, unable to tell whether it is correct, or copied. I want everyone focused on learning, as much as possible.

I told kids the first week that I’d be giving a weekly “Homework Quiz.” These would be designed so that, if you did the homework, you should ace the quiz. Quizzes are given a full class later than the homework is due, so we have a chance to go over the work together and find any misconceptions. I’ve been making the quizzes short (half a page) and only allowing about 15 minutes of classtime. They are quick and easy to grade, and have the tested skills listed at the top. They get entered into the gradebook, divided by skill, under Formative Assessment, which altogether will be 20% of their grade (schoolwide SBG-hybrid policy… long story).

This week, in an attempt to see how things were working (and to keep everyone on their toes) I checked homework. Students were, needless to say, shocked. Three weeks in, and they had already decided I would never check the homework! “I thought you didn’t do that!!” I love teenagers.

Does it work? Yes and no. Here are some gross generalizations based on the first few weeks:

  • Kids who are quick learners either did or didn’t do the homework, and did fine on the quizzes. Honestly, if they could have nailed the assessment just from paying attention in class, do they need homework?
  • Compliant kids – who probably would have done the homework without the quizes – did the homework, but seemed a little clearer on what they should be getting out of it.
  • System-working kids who would do anything to avoid the homework did a minimal amount of the work and often did poorly on the quizzes. I am still hoping that as the weeks go by they will really swallow the connection between practice and success.
  • Kids who lack home support have mostly not done the homework. I suspect that would also be the case in a graded-homework setting. They have done poorly on many parts of the quizes, but not all. The only advantage of this system, for them, is that we have some data on what they know, and what they don’t, so I can get some targeted help headed their way.

There is one other thing about this system that I like: it’s a low-stakes test. Kids get more practice in a testing environment, and another opportunity to recall and build memory.


We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have FUN.
-Paul Lockhart

In theory, theory and practice are the same. In practice, they are not.

-Yogi Berra