Avis foregoes the restroom and expresses some geometry
AVIS was in my Geometry class this year. Or, occasionally he was in my Geometry class. For a few minutes at a time.
Avis is not disruptive in class, as much as he simply leaves class. He’s frequently found just wandering the halls.
He has numerous accommodations to account for his behavior. Some teachers respond with referrals, but I’ve decided little is achieved by repeated referrals when the aim is learning.
In my class Avis asks — insistently — to go to the bathroom. Again and again. On his return it can be minutes before the demand is repeated. If I explain he has just been, he disappears from the room the moment my attention is elsewhere.
And when he is in class, it is hard for him to sit in one place. Avis doesn’t do much geometry, though he will turn in some work if he can sit next to an accommodating pretty girl willing to help.
Earlier in the year the class had used laptops to do various constructions and investigations using Geometer’s Sketchpad. The exercises had not been as successful as hoped, more likely because of the didactic approach I had adopted, making the investigations quite formal.
But Avis’s attention had been grabbed, and the frequency of his bathroom visits slowed. Avis had spent much of his time playing, using the program to draw pictures. As play is a positive start to learning, this seemed a welcome step.
At the end of the year I decided try again with the laptops, using a different program, Geometry Expressions, and adopting a looser teaching style.
Geometry Expressions ¹ is a geometry-algebra system. Algebraic notation is used to “constrain”, or fix, the drawings. The outcomes can also be expressed as algebra. The focus of the mathematical thinking involved is different to that used in attempting to do traditional straight-edge and compass geometric constructions.
If you constrain a triangle side-angle-side, and then try to add a second triangle using the third, unconstrained, side Gx forces some interesting thinking if you try to impose constraints on the second triangle.
Teachers and academics who have worked with Gx readily acknowledge its power for older or higher-level students — my own AP Calc class produced some lovely work based on animating hypocycloids after their exam.
The question has been could the progam help younger or lower-ability children with their mathematical thinking? Avis was not the student I had most in mind when we tried our first exercises using Geometry Expressions.
Students were invited to draw some triangles and then constrain — fix — side lengths, angles, to create right triangles.
The class then had a variety of problems involving Pythagoras. They were asked to solve the problems using Gx to recreate the problem. One particularly interesting task was to find all the possible sides of a right-triangle if two of the sides are lengths 10 and 15.
Below are extracts from the lesson log written as the the lesson progressed (TAG = “talented and gifted”, RR = “rest room”, AA = Avis):
30 mins in, most students able to change sides a and b for numbers + observe how the formula calculates the answer.
Some students use the program to draw pictures not related to the exercises. (This also happened when students introduced to GSP).
40 minutes in… higher level of engagement with activity/focus better than normal for this class. Students interested in confirming answers in discussion.
TAG-type student complains the exercises can be done more quickly without the computer… explain am training him to use the program with aim of using it as a thinking tool + more difficult problems later. Agrees to give it a second go on a more complex problem.
45 mins in… one of most difficult students (usually can’t focus at all, frequent trips to RR, etc) still engaged and asking questions… and has not visited RR! Call this student AA. AA working with girl BB who does not find subect easy, but who tries.
50 mins in… some signs of disengagement among some students.
60 mins in… TAG student happier… still feels he’s working slower, but is seeing the program can aid his thinking.
Two students raise calculation that gives pi/2. Leads to brief discussion that this is 90 degrees (this class not done radians).
65 mins… engagement now down to about one in three students. This is an improvement for this class. Class also much more quiet than normal.
Allow internet access for last 20 minutes… some students still continue with Gx.
Girl BB + boy AA still v engaged and 70 mins in are now trying to work out how to constrain a side using a radical. Still no visit to the RR by AA!
TAG student comments that Gx good “verification tool”, useful for checking answers you’re not sure of.
Pack away 80 mins in.
AA cleans board, still quiet (!!!) and no visit to RR. Wow! AA smiles genuinely brightly when I congratulate him on not going to the RR.
Two days later the same class works on using Gx to solve problems involving the equations of circles. This time Avis is very much the focus of my attentions.
Below are extracts from the log written as the class progressed:
blitz start to lesson… doors locked, etc. Computers eventually arrive from another classroom.
Students specifically told to work in pairs.
13:30 all students working. Enough computers.
AA + BB work together and ask frequent questions (on task).
Questions from students make it v easy to answer with a question focusing on why?
14:48 Class engaged. One TAG student (girl) asked for help, but figured answer while I worked my way round. Questions focus on basic use of Gx… partic how the animation works. These are easily solved questions and students pick up quick.
Talk in room is almost exclusively on the task.
pair work v successful.
13:54 Deep discussion between two TAG students on equations of circles.
Respond to earlier request from BR for help… “I got it! it’s ok…” Doesn’t look up, stays completely focussed.
LI, BB, AA have discussion about where the “variable”s go… these students never talk about “variables”!
14:03 Five students pulled out of class for OAKS state testing… they’re really upset! Includes LI who is working really hard on task.
14:12 AA + BB want to discuss what are the two things on which they must agree for them both to draw the same circle. BB using words like “segment”. AA has not been to the RR!
Two students working in Gx, but drawing pictures. Only occasionally are students caught on the internet.
Easy to get students moving.
Three TAG students now working on the extension activity.
14:12 BB and AA in deep discussion looking at the calculated equation of a circle.
14:22 AA + BB still on task. HT ok writing out definition of circle with help. BB points to an equation and asks “does this make sense to you?” We spot that she had not replaced r for radius with a specific value.
14:24 Tell students to start to shut down… BB shouts out to AA “quickly, let’s do this one.” And they do.
Sadly, departmental decisions about the need to deliver aspects of the curriculum before the end-of-semester test meant an end to the experiments with Geometry Expressions and a return to textbook-based didactics for Avis’s class. At the following lessons Avis mostly left the classroom, though he did frequently ask whether or not the class would be working on the computers.
But the brief experiment did demonstrate that Geometry Expressions can motivate and aid the mathematical thinking of younger and lower-ability children. This was confirmed by similar reactions by lower-ability students in Algebra 2 classes where it was possible to use the technology over a much longer period.
¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project.
Entry filed under: Technology in the classroom, Thoughts from the classroom, What's on the PiFactory blog.... Tags: motivating low-ability students, review of geometry expressions, technology in the math classroom.