Tag Archives: curriculum

Summer of Learning

(This post was mostly written last Tuesday, so dates referenced will be from then.)

So, last summer for me was a summer of travel. Over 10,000 miles round trip, from sea to shining sea, along roads new and old, long and short.

This year will be a summer of travel for my mind, instead. I have a number of different learning projects I am planning/attempting, from workshops to conferences, from in-person classes to online classes.  Nearly all of which is free!

Here are a few of my plans:

Fab Lab

Yesterday, I attended  a class which introduced me to our local Fab Lab at the Community College of Baltimore County (CCBC). A Fab Lab  (Fabrication Lab) is a community-driven and community-accessible location with computers, machines, and other tools needed for making things. A global network of more than 90 Fab Labs worldwide is run out of MIT. Artists, designers, engineers, inventors, as well as ordinary people with an idea they’d like to make  a physical reality, all use Fab Labs.

The Fab Lab at CCBC is a little over a year old. It has a 3D printer, a CNC router, a CNC mill, a laser cutter/engraver, and a vinyl printer. I made the following key chain using the laser engraver, with the help of the lab’s manager who was teaching me how to use the various machines and associated softwares & tools.

Laser Engraved Key Chain

And a sign for my Computer Integrated Manufacturing (CIM) class, done using the CNC router and featuring a picture of a robotic arm:

CIM Sign, milled from medium density fiberboard using the ShopBot at CCBC’s Fab Lab

Very cool.

VEX Robotics & Automation

Yesterday, today, and tomorrow, I’m in a training at the University of Maryland Baltimore County (UMBC, just around the corner from CCBC and its Fab Lab above), where I’m learning more about VEX robotics and their role in the Project Lead the Way engineering curriculum. Including learning how the pieces fit together, the functionality of the various sensors & other pieces, and how to program the VEX kits in RobotC, a variant on the widely-used C computer programming language.

I had two groups of students (6 students total) who began using and programming with VEX/RobotC this year. I learned some of it with them, but really appreciate this chance to work with the automation kits myself and really learn it much more deeply.

It’s been fun so far! Here’s the testbed full of motors, lights, and sensors where we are learning how everything works and how to program:

VEX Test Bed

Tomorrow we’ll be unleashed onto some actual functioning projects!

Online Class(es)

I’ve begun an online computer science course, Algorithms: Design and Analysis I, via Coursera.

I’m planning this summer to learn a lot more about computer science / programming. I only took one CS course in college (CS101). Yet I’ve been somewhat into programming ever since programming the quadratic formula (and many other math-related programs, plus a few fun/game programs) into my graphing calculator in tenth grade. In college, I also used some simple computer programs to design some original fractals (Java) and search for patterns in continued fractions (PARI/GP). And I had many friends in both high school and college who majored in CS or related fields. Since I’ve been teaching engineering, several of the courses I teach have involved programming components (see, e.g., the VEX Robotics and Automation section immediately above).

So I figured I’d like to learn more about CS & programming. I signed up this spring for Coursera’s CS101 class, which (though I can’t find a source for this statistic) I think more than 100,000 people worldwide also took along with me. It included video lecture segments, mini-quizzes embedded into the videos, automatically-graded programming assignments, and discussion fora where students could help one another (since the professor could not interact with so many of us individually). It was a decent review for me, since it’s been years since I took CS101; I learned a few new things including some specifics of the JavaScript language as well as some things about how computer hardware works. Though it was very easy overall.

Coursera, along with a few other recent innovative websites like it, is being referred to as a MOOC: massive online open classroom (or course). Because its classes are free and accessible worldwide (“open”) and are enrolled in by tens or hundreds of thousands of students at a time (“massive”). Some people are talking about MOOCs as the next big step in the educational revolution; I can attest that the experience is much more like an actual class than just viewing lecture videos. I have yet to really engage the discussion fora for help, but I see study groups forming there, both in-person meetings based on geography, and Skype study groups being set up  based on time zone or language spoken. Many other people ask questions in the fora which are quickly answered by fellow students or volunteer teaching assistants.

If this topic intrigues you, check out the two articles linked above (the words ‘some’ and ‘people’). They are quite interesting and thought-provoking about the future of education!

This summer, I signed up for the Algorithms course, which looks like it will be much more challenging, though also like I will learn a lot from it. First I had to pick a programming language. I feel like a lightweight in several languages, from my experience in Java years ago, to knowing a little C based on my robotics teaching experience, to knowing a little Python based on using it to control a virtual robot and help it navigate a maze in an after-school club I advise. I spent this weekend taking a crash course in Python to catch myself up to speed. After that, so far in the Algorithms course, one week in, I’ve programmed a multiplication algorithm and programmed/analyzed the running time of a merge sort algorithm. I’ve spent dozens of hours on it so far, but am really enjoying it!

Both CS101 and Algorithms are taught by Stanford professors; Coursera partners with faculty from several universities.

On a lighter note, I’ve also signed up for this Udacity course that says it will be looking at/analyzing/explaining some cool physics problems, while also visiting actual historical locations in Europe of the scientists who studied them. I’m thinking it will give me some nice perspective and/or new ideas for teaching the physics-related sections of Principles of Engineering (POE).

Materials Science

Speaking of new ideas for teaching POE, I’ve also signed on to take a week-long materials science course at Howard University in Washington, DC. It is being sponsored by ASM International, a materials science/engineering professional society formerly known as the American Society of Metals. They provide free materials camps for teachers across the country at many different sites (see their website for more info).

I signed up for this because a) it’s free; b) it’s local – I can just catch the MARC train from Baltimore into DC; but mostly c) to learn more about and be able to teach the materials unit of POE better. I feel that the materials engineering unit/lessons in POE are often the dullest sections for my students. All of POE is quite difficult/challenging, with a lot of advanced mathematics and high-level physics concepts. But the other units I am able to better balance out between the difficulty of the concepts and the exciting projects we do. In this unit, students analyze properties of various materials, discuss what causes those properties, discuss how materials are used in manufacturing processes, do various materials-related math word problems, and use a stress analyzer machine to pull apart (stretch it until it breaks, called a tensile test) a piece of metal and then analyze its graph. While students love seeing the metal piece snap in two, I am not able to sustain that interest through the rest of the unit, which I take as a failing on my part. So, I hope to learn more during my week of Materials Mania, as well as to find ways of engaging students better in the topic.

Fullerene Nano Gears, image from Wikipedia

Hooray for the start to my summer of learning!

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Golden Ratio

I can’t believe I’ve taught math for almost six years and not done much with the golden ratio, phi (φ) ≈ 1.6180339887, and the associated Fibonacci sequence 0,1,1,2,3,5,8,13,… (add the previous two numbers to get the next number, so 8+13=21 c0mes next)! Especially when so much of my own research in college was connected to phi. You see that spiral up in the blog heading? It’s related to the more famous golden spiral and Fibonacci spiral,

Fibonacci Spiral

but it’s actually something never-before-seen I discovered about six years ago: the Yates Golden Diophantine Spiral. It has the special property that, when centered in a coordinate plane at (φ,0), it has x-intercepts at precisely ratios of consecutive Fibonacci numbers! For example, at 3/2, then at 5/3, then at 8/5, etc. That’s something those other spirals can’t claim!

And then there’s my research into continued fraction representations of irrational numbers, of which phi is the simplest:

It turns out that every quadratic irrational –like φ = (1+√5)/2– has a continued fraction that repeats periodically at some point (e.g. √7 = [2;1,1,4,1,1,4,1,1,4,…] has period length of three because 1,1,4 repeats). Now this by itself is pretty cool since quadratic irrationals’ decimal expansions continue forever but never repeat! Anyway, seven years ago during the summer, a group of three other undergraduate students, myself, and our faculty advisor were able to find a way to write alternative continued fractions for every quadratic irrational number with just a single number repeating every time (period length one). This is true even when the standard continued fraction has a period a million numbers long, which, when you think of it, is pretty surprising! Our results were published in the Journal of Number Theory. In fact, our first step toward discovering these results was experimenting with some Fibonacci identities like these.

So, it seems like a natural topic for me to include phi and Fibonacci in my classes, especially with all the geometry involved in golden rectangles and spirals, and the quadratic equation that generates the algebraic number phi. But, aside from a short detour my first year teaching Algebra I, where we were talking about patterns including the Fibonacci sequence, and my students asked me to explain my research, I have not done anything with phi in teaching Geometry or Algebra 2.

That’s all about to change. I was inspired by the following video (click through to see parts 2 and 3 as well):

I’ve developed a project centered around that video, that connects Algebra 2 to Biology and Art. More on that tomorrow. 🙂

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November in Geometry

Although this is my sixth year teaching, I’ve been struggling with classroom management issues this fall in my last-period Geometry class. So we haven’t been able to do some of the cool projects I talked about last year (click the Geometry tag to see more). And a few topics we haven’t been able to delve into at quite the same level as I could with a more-motivated group of students.

A few details on some topics we’ve worked on during the past month or so:

The lesson on three-dimensional polyhedra went fairly well for the first two parts. Students constructed polyhedra from nets and by building their skeletons out of gumdrops (vertices) and toothpicks (edges). They discovered the relationship between vertices, edges, and faces found by Euler (V+F=E+2). But when I tried to bring the whole class to proving that only five regular polyhedra exist, I lost 80% of the class. I don’t know if it was too many steps, too long for their attention spans, an aversion to the logic of proofs, or the overall class dynamic. I don’t believe the math was too complicated for them (it just has to do with angles in regular polygons, spatial relationships, and our previous topic – tilings of the plane). I provided a sheet for taking guided notes. But much of the class turned that sheet in without having taken any notes.

Some of the more successful lessons have been a few that tied into what my students were learning in their engineering class. In late October – early November, my sophomore Geometry students were building and analyzing truss bridges in their Principles of Engineering course. Several teachers got together to plan lessons in various subjects that tie into the topic of bridges. In October, near the beginning of that unit, I did a lesson on the strength of various shapes. Students tried to use paper to hold the most books at least one inch off the table. They constructed a bridge that could hold the most rolls of pennies, using just one index card. And another bridge of multiple index cards, designed for length.

A couple weeks later, after they had developed designs in the engineering class, I had them analyze some of the geometry of triangles. This connected their bridges to what we were currently talking about in Geometry, with triangle congruence, proof, naming, and the Pythagorean Theorem.

Bridge Geometryhttp://www.scribd.com/embeds/73632593/content?start_page=1&view_mode=list&access_key=key-1xgwxwz3mcqb0iig9brt//

We’ve also learned about isosceles triangles, angle relationships, and circles in the past month.

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Reminders: Please support my moustache & Baltimore students by donating, and please support my partners in Baltimore’s NaBloPoMo by visiting and commenting:

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November in CIM

Within the past month (stretching back a bit into October), we’ve been working on two major projects:

CNC Programming and Mill Use

CNC-Milled Initials, Fall 2011

To complete this project, students first designed their initials (out of straight lines and arcs), plotted points’ coordinates, learned how to operate the CNC Mill properly and safely, programmed in G&M Code, troubleshot their problems via a code verification software, and then milled their blocks.

Robotic Arm Programming & Use

Robotic Arm, Stacking Three Blocks, Fall 2011 - Could still use a little adjustment on block alignment

For the robotic arm, students work through a scaffolded sequence of activities: from programming the arm to pick up and move a block, to the difference between recording and teaching positions, to stacking three cylinders, to stacking three cubes using trigonometrically-calculated roll angles, to moving the robot linearly vs. circularly vs. by joint, to using variables and loops, to using inputs and outputs, to using variables and loops and subroutines and inputs and outputs simultaneously. In addition, we look at the history of robotics and automation, and learn the parts and types of robotic arms.

Robotic Arm, Circular & Linear Motion, Spelling Out Letters - It was very difficult to get the right grip on the marker so it was the right height and didn't angle itself upon contact with the paper

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Reminders: Please support my moustache & Baltimore students by donating, and please support my partners in Baltimore’s NaBloPoMo by visiting and commenting:

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Trip to BGE

Yesterday we took a tour of Baltimore Gas & Electric‘s Spring Gardens facility in South Baltimore.

We heard about the environmental protection efforts BGE has been implementing at that site. We learned about the history of BGE, back almost 200 years to its beginnings as a gas light company. We got to listen in on calls being handled by customer service agents (!). And then we discussed and toured their field of 572 solar panels.

Solar Panel Array at BGE Spring Gardens

We talked about how the solar panels worked, how much energy they generated, how it was turned into electricity used by the other buildings there, and how it was helping the environment.

This was awesome because our tenth grade students (the target audience of this trip) had just been learning about energy, power, and specifically renewable energy sources. They had recently built cars powered by solar panels, and also had created alternative-energy-fueled model power plants that distributed energy through circuits to model homes and businesses. I love the chance to make connections, both between classes and from class to the outside world.

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Trigonometry with Algebra 2

Since this May may be my last month teaching math (for a while? forever? I doubt my departure from math will be permanent … more on the story behind this later), I thought I’d get into the swing of things and connect back into the theme of my blog by declaring that May is Maryland’s Math Madness Month! [To give due credit: this motto was originally thought up by my father as a way I could get kids excited about taking the Algebra HSAs, coming up now in less than two weeks.]

To kick off the month of math madness, I wish to ask other teachers of Algebra 2 with Trigonometry (A2T): how do you balance covering all of a (full-year or full-semester) Algebra 2 course in a reduced time-frame so that there is room in your course also for Trig?

A2T Collage

I suppose there are two main options for dealing with this addition of new material into an already full course: leaving out pieces of content, or teaching all the content at a faster pace.

This is something I think about every year I guess, but even moreso recently when helping students with their Algebra 2 Twilight make-up coursework and having discussions with the Precalculus teacher about what he will expect my students to bring with them to that class.

The Twilight students had question after question to answer about completing the square, which is a method I don’t cover when teaching A2T. For me, solving quadratics by factoring, “doing the opposite”, and using the quadratic formula is exhausting enough, both in the sense that it exhausts the techniques required to solve quadratics of every form, and in that my students are tired of so many methods without adding a fourth. My students have been know to complain that they are learning something new every day in my class (!).  Am I wrong for leaving out this method? Should I perhaps teach completing the square instead of the quadratic formula since it shows deeper understanding of the math involved? I don’t have enough time to do both (plus the other two methods I mentioned, which are even more fundamental).

Similarly, I treat complex numbers very lightly (last year, with the 9++ snow days, I even skipped them!). And I hear on the web about some Algebra 2 teachers teaching rational functions, which I never even conceived of as an Algebra 2 topic, since gaining an abiding understanding of polynomials is challenge enough.

So I guess that lands me primarily on the side of leaving out content. A faster-paced curriculum would leave more students lost, and I do not have selection criteria for entering the class as some teachers might. Additionally, this relates to my philosophy of math teaching, that it’s better to learn fewer things deeply than to shallowly cover everything. I try to focus my attention on the things that connect A2T to prior math and future math (e.g. “doing the opposite” as equation-solving technique and function transformations), that connect it to other subjects, that engage students with project-based learning, and that highlight big picture concepts and skills.

But it’s still a struggle, and I doubt myself (maybe I really should be teaching completing the square; maybe conic sections are more important than my Olympics research project and should replace it in my choice of topics). Especially since that Precalc teacher is counting on me to teach them certain things they will need when they arrive in that class (and just as the Calculus professor is counting on Precalc teachers to cover certain key topics).

So, A2T teachers, how do you deal with the pressure? Do you teach at hyperspeed, or what topics do you cut? [Other teachers feel free to weigh in too 🙂 ]

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Principles of Engineering Skills

So I’m thinking of giving a skills list a try in my Principles of Engineering (POE) course this semester too. It wouldn’t be full-on standards-based grading (SBG), since in such a project-driven class I need projects and reports to be the major component to student grades, but I think I can adapt a skills list for quizzes (and occasional outside-of-quiz skills demonstration). I’m tentatively planning 50% projects, 25% skills quizzes, and 25% portfolio–including engineer’s notebook.

Below find my first draft for a skills-based outline of how I intend to teach POE this semester. I will be teaching a mix of ninth and twelfth graders, with backgrounds ranging from Algebra I to Precalculus. This will be tough, as POE is the most math-intensive of the PLTW engineering courses. I must try to teach advanced math applications while not boring my students out of their minds, while at the same time exposing students to the great concepts and societal role of engineering and its subfields.

To all the former, current, and future engineers out there, I welcome your input on the skills listed below. Likewise to engineering high school teachers across the country (for your reference, I’ve mixed up the unit order due to equipment lacks: my order goes Unit 2,4,3,1). Or anyone else with an opinion on engineering education.

While I am constrained somewhat by PLTW’s POE curriculum, I do have some choice in what I emphasize and in which skills I test. Are these skills phrased well? Are they representative of what engineering is all about? Are the major subfields of engineering represented (this is a survey course)? Are there any that are too vague (or too narrow) or seem like they don’t belong?

I’m still welcoming feedback to my Algebra 2 with Trigonometry skills list here for about one more day, so my more mathematically-minded readers may like to head over there to ponder and critique.

POE Skills List

General STEM Skills

  1. Solve equations for a single variable
  2. Substitute numbers for variables in algebraic formulae
  3. Measure lengths and angles to appropriate precision (given the context of the application and the accuracy of the tool)
  4. Use trigonometry to solve for missing sides or angles
  5. Apply the Pythagorean Theorem to find unknown sides in right triangles
  6. Use trigonometry to find unknown sides & angles in right triangles
  7. Use the digital dropbox on TS3/Blackboard to submit work
  8. Identify problems to be solved in an engineering context
  9. List multiple possible solutions to engineering problems
  10. Evaluate each possible solution based on specifications & test results
  11. Show knowledge of, and skillful application of, the engineering design process
  12. Show knowledge of various careers in engineering and other STEM fields

Unit 2 – Materials & Structures

  1. Identify five types of bridges by name, definition, and/or picture
  2. Split a force vector into its x- and y-components
  3. Calculate the centroid of various shapes
  4. Calculate forces and moments acting on various objects
  5. Pick appropriate formulae relating to stress, strain, and material testing
  6. Analyze stress-strain graphs to determine material properties
  7. Calculate bridge efficiency
  8. Analyze a bridge for structural and material strengths and weaknesses

Unit 4 – Statistics & Kinematics

  1. Collect and analyze data using statistical measures of center and variance
  2. Calculate speed and velocity
  3. Calculate the effect of gravity on velocity and position
  4. Analyze horizontal and vertical components of projectile motion

Unit 3 – Control Systems

  1. Create flow charts to represent a process
  2. Identify inputs and outputs in a control system
  3. Identify elements of a flow chart or RoboPro program and their key attributes
  4. Utilize branches in a flow chart or RoboPro program
  5. Utilize variables in a flow chart or RoboPro program
  6. Interpret a flow chart or RoboPro program
  7. Identify open and closed loop systems
  8. Demonstrate an understanding of pneumatic and hydraulic power

Unit 1 – Energy & Power

  1. Calculate ideal mechanical advantage for each simple machine
  2. Calculate actual mechanical advantage and efficiency for each simple machine
  3. Calculate gear ratio
  4. Demonstrate an understanding of electricity and electrical circuits (series/parallel)
  5. Use Ohm’s Law and Kirchhoff’s Laws to calculate resistance, current, and voltage
  6. Calculate work, energy, power, and power efficiency
  7. Demonstrate an understanding of the laws of thermodynamics and thermal energy transfer
  8. Demonstrate knowledge of alternative and renewable energy sources

A total of 40 skills. What do you think?

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