Tag Archives: paperless


Happy Thanksgiving Eve!

Marble Sorter

Marble Sorter

I take photos all the time in my engineering classes. I hang them up on the wall, post them to our school website, and have my students use them in their projects. To me, a photo is an excellent documentation of work done. For example, the photos below of initials carved by a CNC mill prove to me that these two students successfully applied mathematical knowledge of Cartesian coordinates, engineering knowledge of G&M Codes (the programming language used to communicate with a CNC mill), and technical knowledge of how to operate the CNC Mill machinery, to produce the final product.

Or take this photo of a robotic arm that has just successfully built a pyramid out of wax blocks. It shows me that a student has mastered the skills of how to control the robotic arm, how to record positions, and how to program the arm to perform that task. Note that I do observe the student running the program before the photo is taken, so it can’t be staged!

Robotic Arm with Pyramid

Robotic Arm with Pyramid

Now, I am less accustomed to taking and using photos in my math classes. But I would like to move more in that direction. I’d like to use photos in math class on as regular a basis as I do for engineering, plus be able to hang those photos up and celebrate my students’ work. I took photos this year of our class’s huge Sierpinski Triangle Print, and of some 3-dimensional polyhedra that my geometry students built. What else can I take pictures of? How else can I use photographs (or even videos) in my math class, not to introduce a lesson but as part of the projects that we do?

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Filed under engineering, math, teaching

Shadows, Mirrors, Scale Models, & School Measurement

I’ve been itching to do this for days, but clouds and rain got in the way.

The past two weeks have been overcast and have had rain in the forecast nearly every day, so I was excited yesterday when at lunchtime I saw a break in the clouds, sun in the sky, and shadows on the ground! We would take a mini field trip outdoors and find the height of our school! This would be done by measuring shadows, drawing similar triangles (one from the school, its shadow, and the line of the sun’s rays connecting the top of the school to the furthest reach of its shadow; the second triangle from a pole, its shadow, and the sun’s rays, the pole being an item we could measure both height and shadow length directly), setting up a proportion, and solving for the unknown school height.

After lunch we began class with a warm-up challenge, then I posed the question all WCYDWT style to pique student interest: How tall is Patterson High School? I tried to ‘be less helpful’ as students threw out a variety of ideas of how we could figure it out (measure the height of my classroom and multiply times three for three floors, climb up to the top of the school with a ladder and tape measure, etc.). After some discussion, I turned out the lights and used a flashlight to cast a shadow on the floor. We talked about how shadows are formed and drew pictures on the board.

Unfortunately, as we stepped outdoors the sun was disappearing behind a cloud! Disheartened, we went back indoors and wrapped up construction of scale models of various classrooms in the school.

Last night, saddened by my shadowless plight, a teacher on Twitter suggested a cloudy-day alternative (or any-day comparative addition) of using a mirror to create similar triangles. When you can see that top edge of the building in a mirror laid on the ground, your line of sight to the mirror is reflected at an equal angle up to the top of the school. Add in yourself, the building, and the distances from the mirror out to both, and you’ve got similar triangles! The funny thing is that, even though I have practice questions in the project about mirrors and similar triangles, it never occurred to me to actually get a mirror and do this. Perhaps because my main experience with mirrors is as vertically affixed to a wall instead of portable and horizontal? So, in any case, this morning I bought a $2 compact mirror at the grocery store.

There’s a happy ending though! Today the weather was beautiful, so we made the trek outdoors again. Students and faculty stared as we walked through the halls, tape measure, pole, yardsticks, mirror, and notebook in hand. We got outside and set to measuring/recording. A teacher on the second floor leaned out of his window, said hello to the students, and asked one of them to toss up a chalkboard-eraser that had fallen out. We measured the school’s shadow, the height and shadow length of the pole (students knew how this would yield similar triangles due to our talk yesterday). Then students took turns moving back and forth until they could see the top edge of our school in the mirror on the ground, and then we measured one student’s eye height, the distance from that student to the mirror, & the distance from the mirror to the school. Since we hadn’t discussed this in advance, they weren’t sure exactly how these mirror measurements would fit in to help figure out the school’s height.

We raced back indoors since the end-of-day-announcements would be coming on in only a few minutes. The students drew similar triangles all over the board, figuring out how the mirror would be used as well as the shadows.

Mirror-Caused Similar Triangles and Calculation

Mirror-Caused Similar Triangles and Calculation

Using proportions and coming up with different answers for the school’s height, we discussed sources of error in measurements, and brought the lesson to a conclusion just as the loudspeaker came on!

All in all, an exciting day and a reinforcement for why I believe in project-based learning. Perhaps the ‘revolutionary’ new Baltimore teachers’ contract will tie our evaluations and pay to how many of our students can determine the heights of unknown buildings using only a few tools and the power of math, rather than just looking at standardized test scores? Well, I can dream, can’t I? 😉


Filed under math, teaching

Paperless/Lesspaper teaching

Back in August, I set myself a goal of nearly paperless teaching.  For several reasons, I have not been able to go completely paperless, but I am certainly using less paper this year than previously.

I still usually print out quizzes and tests, though I have made forays into online quizzes.  These have been difficult to make in the school’s learning management software, Blackboard, while other formats that other teachers around the country have used successfully (e.g. Google Forms) have been blocked by the school’s network.  In addition, I print out photos of student work and their reports/presentations, for inclusion in students’ portfolios.  My students also maintain an online portfolio, but are required to have a binder portfolio for their engineering courses.  I also sometimes still print out homework assignments, since I believe that students who have a piece of paper in their hands are more likely to complete the assignment.

While ‘paperless’ would be an inaccurate descriptor of my teaching this year, ‘lesspaper’ is an invented adjective I can absolutely apply.  I am printing and copying much less than last year.  I also make sure to direct students to recycle rather than trash any paper they do use up.

For Earth Day last week, I pledged with 1500 other teachers worldwide to teach completely without paper for a day.  I had limited success in that I taught my two courses without paper (caveats below my description of how I taught paperlessly)!

  • In Principles of Engineering (POE), Thursday is quiz day.  Since migrating a quiz to Blackboard has been difficult, I kept the quiz in its original Word document format.  I placed these digital documents before class onto each student’s computer.  When the students arrived, I asked them to eliminate wrong answers from multiple choice questions and type answers to open-ended questions directly in the document.  They then saved the document with their names and submitted them electronically by e-mail or through the ‘digital dropbox’ on Blackboard.
  • After completing the quizzes, POE students worked (without paper) on building and programming their marble sorters.
  • In Algebra II with Trigonometry (A2T), we began with each student completing a different warm-up challenge of solving a trigonometric equation–I posted twelve different equations on the classroom chalkboard, the classroom dry-erase board, and a few individual dry-erase boards.  This was a nice change of pace, students standing up and moving around the room at the start of the period.  They also liked that each student had his own question to answer.
  • After the warm-up challenge, my A2T students investigated transformations of sine and cosine waves, with an online GeoGebra applet and an adaptation of the activity described here by Jessica (with thanks also to David).  They completed the assignment and turned it in by e-mail/dropbox.

Now for the limitations, due to encountering the unexpected realities of daily teaching:

An advisory session with my homeroom students, normally scheduled for Wednesdays, was at the last minute postponed to Thursday (Earth Day). I had already photocopied the standardized lesson on Wednesday before being told of the postponement, so I went ahead and used those papers to guide my advisory meeting. It did not seem worthwhile to waste the already-copied papers by placing them directly into the recycling bag, in order to come up with a new non-paper activity. The same day, another teacher was out sick, and he had left a paper assignment for his class, which I watched for one period. So I handed out that paper to his students.  Finally, a student cut my class that day, and our school only accepts hard-copies of cut slips or other behavior referrals, so I did print out a cut slip for that student.

As you can see, it’s difficult to cut out paper entirely for even one day!  But I am happy about the successes I have had in lessening paper use in my classes, for Earth Day and throughout the year.

If you’re interested in the idea, more ideas for teaching paperless lessons can be found on the Teach Paperless Blog, the Paperless Earth Day Wiki, and the Going Paperless Google Doc.


Filed under teaching

Vector Video

I was thinking recently about how best to teach trigonometry, and especially how to teach the long and difficult processes of vector analysis and truss calculation.  Now, it seems to me that these processes aren’t really going to sink in unless a student practices them a number of times. But that sort of independent practice is difficult to achieve for such complex, many-step processes.

Now, I can do guided practice of these processes by letting students lead me through a 20-minute example on the board.  But then the student gets little individual time to work it out herself.  If I try to have the students work a problem on their own too early, I’ll be called in fifteen directions at once as each student gets stuck in a different place and needs one-on-one help.  To varying degrees of success, I may try to scaffold a problem by doing some parts for a student or asking him leading questions; or I may list the steps in the algorithm on the front screen while students work on applying it.

However, to get the students all the practice they need to get comfortable with these topics, they do need to work on them in homework too.  And I cannot be there to scaffold the instruction; I can provide neither one-on-one assistance nor lead a classroom guided practice.

In an effort to help students with their homework in these areas, I decided to videotape myself working through a vectors example:  that way they could see the process while working on a homework assignment, with the ability to pause to work on a similar part in their own example, and the ability to rewind to hear a step over again.

I’d appreciate any feedback, and especially constructive criticism, as I’d like to make videos a recurring part of my teaching this year.  I intend to make a truss calculations video quite soon, to follow up on these ideas.

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Filed under engineering, math, teaching

We wish you a merry August and a happy new year!

Hello, world!  A new blog; a new school year; a new beginning!

Today here in Maryland, students begin a new year of school.  Teachers (like me) started last week, and have had a week full of meetings:  department meetings, academy meetings, district-wide meetings across town in an empty school.  Plus one day to set up our classrooms for the new school year!

As I write to you, I am beginning my fourth year teaching here in Baltimore, where I teach math and engineering at a large public high school.  Although I have greatly enjoyed learning and then teaching engineering each year, my first love is mathematics, and it is math and math education that I wish to be the main foci of this blog.

As my inaugural post, I wish to meet Sam Shah’s challenge and declare a triplet of New Year’s Resolutions that I have set myself:

  1. In the coming school year, I shall strive to be nearly paperless as a teacher.  Inspired by another Maryland blogger, I wish both to reduce my environmental impact* and to encourage use of the tools of technology my students will draw upon as they enter the 21st century world beyond school.  While I plan on providing a paper syllabus the first day of class (together with a whole host of forms that must be signed by students and parents), from that point on I shall post all materials, quizzes, projects, homeworks, and grading criteria online on the course website.  I shall encourage but not require that my students likewise manipulate and submit their work by computer instead of in hard copy.  The only things I shall print out are color copies of students’ technical reports and project presentations, so that these may go in the students’ portfolios.  I am lucky enough to teach in a computer laboratory, which makes a goal such as this one possible.
  2. In the coming school year, I wish to implement a more structured homework regimen.  In the past, I gave homework every night, but it was haphazard:  p.238 #1-3,5,8-10 one night, a worksheet I developed the next; practice with calculating bridge efficiencies, alternating with completing the introduction of a technical report so that it may be proofread the next day.  With more structure and pattern to the homework assignments, I hope students will fall into a rhythm with the week’s assignments and have greater rates of homework participation and completion.  As part of this goal, every daily practice home assignment should be designed by me and consist of sections involving basic, moderate, and advanced application of a skill.  Moreover, these daily practice homeworks will alternate with a weekly homework structure as follows (for engineering classes):
    • Vocabulary weeks will alternate with engineering article weeks.
    • In a vocabulary week, the vocabulary will be assigned Mondays.  Definitions must be provided in hand written form by Tuesday.  On Thursdays, students will study for a vocabulary test that will take place the first ten minutes on Fridays.  Tuesday night and Wednesday night will include daily practice style homework.
    • In an article week, students will be assigned an article to read, either on engineering current events, or on a subfield of the engineering profession.  By Wednesday, students should have posted answers to both factual and opinion questions in an online discussion forum.  By Friday, students should have responded to two others’ posts to create an interactive article discussion.  On Monday and Wednesday nights, daily practice style homework will be assigned.
  3. As my third and final resolution, in the coming school year I hope to make and post videos.  This resolution is my most vague.  I might post videos of me explaining a math concept or procedure, so that students have access to this explanation at home while they work on homework.  Perhaps I could videotape students explaining a procedure too.  And/or I might post videos of student work.  Toward this goal, I welcome advice from those who have worked with video before (I have not).

Thanks for reading!  And please let me know your thoughts, especially on how best to implement videos in class.

* I can remember getting into school at 6:30am, the first teacher there, turning on the copier and making thick, many-paged packets for my engineering classes.


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