Teaching Geometry at Mountain View High School*: Technology Integration

It is 8:00 AM and a few 9th grade students slowly enter the classroom. Music is playing, ballads and songs from an earlier decade—I recognized “Hotel California.” Brendan Dilloughery , working on his laptop at his desk, welcomes each student by name and they sit in their pod of desks, some slowly unpacking their notebook and laptop or tablet (this is a Bring-Your- Own-Device district**) from their backpacks. Other students put in earbuds to listen to their favorite music or program, and a few   just stare into space.

A veteran teacher of nearly a decade in international schools in Ecuador, Switzerland, and other places, Dilloughery is in his second year at Mountain View. He teaches geometry and computer science. Tall, energetic—constantly on the move even before the buzzer sounds for the geometry 1 class to begin—the teacher has the agenda for the lesson on the white board. Trimmed beard, mustache, and goatee, the teacher is wearing a maroon long sleeved shirt and dark slacks. He gazes around the room seeing pods of 3-4 desks scattered across the medium sized classroom slowly filling with students.

The buzzer sounds at 8:10 (the period will end at 8:55) and Dilloughery gets the 19 students’ attention. He asks them to take out their homework—three students sitting near me tear out a written page from their notebook —and tells the class that he will come around and stamp their homework (the stamp is a large checkmark). Dilloughery walks around as students place their homework next to their laptop or tablet which they open and go to Google Classroom where they access the homework assignments and geometry proofs for the day (all students have a textbook at home from which the teacher assigns homework).

The “agenda” for the day is on the whiteboard:


–Blue Angels tomorrow at lunch

–Review homework

–Proofs -big picture

–IXL-C 8

BD clssrm.jpg

After stamping homework, the teacher asks students to close their lids at a 45 degree angle (after all, it is geometry, I think to myself). Students do. At the front whiteboard, Dilloughery then proceeds to go over step-by-step a problem that requires a logical proof. Students are encountering proofs for the first time in the course and the teacher is both explaining the steps and giving them practice. On the whiteboard is the following:

2.6 Prove Statements about segments and angles

Prove that the distance from the restaurant to the movie theater is the same as the distance from the cafe to the dry cleaners

Givens: TS-CF


Prove ?

Teacher goes over each part, interspersing his explanation with questions for students (“what was the postulate from yesterday?” “Why is this last statement transitive property?”). He calls on students by name. After finishing, he says:

“Now, we are going over the homework. What questions do you have from your homework?”

Students call out three problems from text that they had to do for homework; teacher jots down the numbers and puts up the homework problems on the screen. In a question-and-answer format with class, Dilloughery goes over each of the problems students asked for help.

I look around the class and all students appear to be listening or taking notes. No one I can see is obviously off task, that is, looking at computer screen or cellphone.

In breaking down each problem into parts and getting at concept of congruence in a proof, the teacher dramatizes what he is doing by stretching out arms, bending legs, making side comments to the class, and moving around the front of the room. The class seems used to this kind of teacherly enthusiasm since some students smile and others watch carefully what he does at the whiteboard. ***

He moves to the other problems that a few students said were hard for them. They are two-column proofs.

BD WB problem.jpg

Teacher calls on student: “what am I going to write for step 1?” Student answers correctly. Dilloughery then goes to next step and says this could be a postulate involving angles and adding angles together. “What would that be?,” he asks. One student answers and the teacher, in a positive burst of happiness at the answer, says “Oooh.” Then he acts out the answer by taking a long step forward on the floor in front of the whiteboard. Students around me break out in smiles.

Dilloughery walks the class through the other problems that students had raised with homework. He encourages members of the class to call out answers—usually he names students when he calls on them—as he finishes this portion of the lesson.

Teacher then segues to next and final activity. He directs students to begin practicing with a partner two-column proofs on IXL, an online math software program that the teacher uses for geometry.

Holding blue note cards with student names in his hands, he shuffles the deck of cards and makes up pairs randomly. He comments on who the partners are going to be, pointing out their strengths. Some students laugh. Because the names are paired randomly, students take their tablets and move to different pods in the room to sit with their partner. Dilloughery announces that partners will spend 15 minutes on the two-column proof. He turns on the music and it plays softly.

After a few minutes, I look at three pods near me and see pairs of students are looking at one screen and discussing what they need to do to complete the two- column proof..

While students are working on the task, the teacher moves from pair to pair asking questions, looking at their screens to see what the partners have typed. He has a comment for each pair. For the entire activity, Dilloughery moves swiftly from pair to pair seldom stopping more than a minute or two as he quizzes the partners and listens to their answers. For one set of partners, I hear the teacher say, “you got it” complimenting them on their work. At another pod of four desks, two pairs are having some problems. Dilloughery stops there and goes over what the students have done, asks for their reasoning, raises questions, listens to student answers, and points out glitches in partners’ reasoning—all in a few minutes before moving to another pair.

Teacher alerts students that the bell will ring in less than a minute and they should pack up. Buzzer then sounds and students  slowly leave the room. These 9th graders are finished with geometry for the day.


* Part of the Mountain View-Los Altos High School District, Mountain View High School has  just over 1800 students (2015) and its demography is mostly minority (in percentages, Asian 26, Latino 21,  African American 2, multiracial 2, and 47 white). The percentage of students eligible for free-and-reduced price lunches (the poverty indicator) is 18 percent. Eleven percent of students are learning disabled and just over 10 percent of students are English language learners.

Academically, 94 percent of the students graduate high school and nearly all enter higher education. The school offers 35 Honors and Advanced Placement (AP) courses across the curriculum. Of those students taking AP courses, 84 percent have gotten 3 or higher, the benchmark for getting college credit. The school earned the distinction of California Distinguished High School in 1994 and 2003. In 200 and 2013, MVHS received a full 6-year accreditation from the Western Association of Schools and Colleges (WASC). Newsweek ranks MVHS among the top 1% of high schools nationwide. The gap in achievement between minorities and white remains large, however, and has not shrunk in recent years. The per-pupil expenditure at the high school is just under $15,000 (2014). Statistics come from here and mvhs_sarc_15_16

**BYOD began two years ago in the District.

*** Dilloughery told me that his principal joshed him by saying, “I think I could plug into your enthusiasm and run for a couple of days.”

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Filed under dilemmas of teaching, how teachers teach, technology use

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