Brainstorming an Integrated Unit: Projectile Motion

As we go through our Year of Planning to become an Academy of Engineering, we are working on plans to try out an integrated unit. An integrated unit is where students are learning about a common topic across all their classes. The topic needs to be rich enough that it can tie together parts of the curriculum from math, science, literature, language arts, social studies, and engineering.

This idea is not new, and it has gone by other names (“interdisciplinary learning” among them). On a smaller scale, most of the engineering projects that I teach in the Project Lead the Way curriculum bring together a large subset of these subjects. For example, our Introduction to Manufacturing Project in CIM brought together writing and presenting skills (English), research into topics in the history of manufacturing (social studies), calculating costs and striving for efficiency (math), as well as constructing and programming an elevator that can be called to any of three floors (engineering).  But that was all in one course, instead of bringing in all those different classes. Our collaborative art-geometry fractals project is another example of small-scale building of units in different courses that spiral together around a common theme.

CIM Freight Elevator

Next year, as part of the Academy of Engineering model, we may have interdisciplinary teams of teachers who teach the same groups (“cohorts”) of students. That will make integrated unit planning much easier, since the teachers on the team can get together during a common planning period, discuss the concepts, skills, and standards that are coming up in each course, and tailor the integrated unit to address those concepts/skills/standards for that particular group of students. Timewise, that will allow integrated units to be done more frequently (monthly?). And students will get to see connections between disparate subjects when the same topic/project is driving instruction in all four of their classes! This year, though, since we don’t have teachers who teach the same groups of students, we want to give it a try by having all teachers in our academy integrate a common theme into their lessons, sometime between now and the end of the semester. And we hope to try again in spring, improving our implementation of integrated instruction in the interim.

One possibility for our late-autumn integrated unit is the Bombs Away unit, developed by ConnectEd and posted on the NAF website (login required to see NAF curriculum). It takes a lesson on catapults and projectile motion, from the Principles of Engineering course, and ties it together with other academic subjects. For example: discussing parabolas in algebra, while learning about bombings in World War II in history, and debating the morality of bombings in English class. However, not all subjects have a lesson plan in the integrated unit (it was developed for California 10th grade courses, I believe).

So I hoped to do some brainstorming here with the help of my Personal Learning Network. Listed below are the courses being taught in our academy this semester (I hope I haven’t left any out). My goal is to get one or more lesson ideas for each subject that will connect to projectile motion or bombing. Can you suggest one or two?

• 10th grade English
• Chemistry
• American Government
• Geometry
• 11th grade English
• Biology
• World History
• Algebra II
• Spanish
• 12th grade English
• Precalculus
• Principles of Engineering
• Computer Integrated Manufacturing
• Digital Electronics
• Design Technology / CAD
• Leadership Education / JROTC
• Art
• Psychology
• Economics
• African-American Literature
• Creative Writing
• Robotics

Mathematical Proof

Proof is one of the most important concepts in mathematics. Proof separates math, where statements must be rigorously proven using deductive reasoning, from science, where statements of accepted truth are induced from observation of repeated trials. This is part of what makes Gauss claim that mathematics is the “queen of the sciences” and why mathematicians sometimes take a snooty attitude toward scientists, whose truth could be overturned at any moment by counter-evidence.

So how do we (and how should we) teach proofs? At the high school level, proof is usually only taught in Geometry, then ignored both before and after in the Algebra-Calculus sequence. And I’ve railed before at how proof is taught in geometry as a dulled-down, poorly-motivated, memorize-these-common-reasons-to-go-in-the-second-column-that-prove-things-which-are-already-obvious-anyway method, whose seeming pointlessness turns many students off to math even more than they had been before.

Irrational Cube ponders this question in a post on “Proof (1st day solo teaching)”:

Here lies the problem.  Proofs in their final form are a series of steps from givens to a conclusion.  However, that is not how people go about proving something.  A proof requires thought of how to get from where you are to where you want to go.  You feel around in the dark, think you’re getting somewhere but hit a dead end, give up, try and prove the opposite, give up again, start over, discuss it with friends, play with different ideas, sleep on it, draw a diagram, turn your diagram into a picture of a dragon eating the proof, give up again, wait a week, have a brilliant idea and then write down your proof.  When we ignore all the intermediary steps then the question of “why are we substituting that” and “how do you know we’re supposed to do that” become unanswered.  We are doing that because everything else that we didn’t try wouldn’t have worked. Of course, you can’t do all those things in a one hour period, so you skip to the end.  In doing so, you deny the essence of proofs and equate them to unquestioningly following directions.  No wonder students dislike proofs – they get the wrong idea about them.

…

What am I missing here.  There has to be another way.

In my geometry class, I will be walking students through a proof that there are only five regular polyhedra (the Platonic solids) next week. I don’t expect them to come up with it on their own, but plan to do it as a whole-class activity, asking questions and having the students’ answers lead the proof along. I’m hoping to make sure the students want to know the answer before we delve in (there are infinitely many regular polygons, so many that we stopped naming them after a while and just call them things like 23-gon, so how/why should there only be five regular polyhedra?!?). I’m also excited because the proof connects back to our work with regular tilings of the plane, which in some sense are simply degenerate regular polyhedra, where the vertex angular sums didn’t allow the faces to fold up into a 3D shape.

On the whole, I try to encourage my students to notice patterns and make conjectures based on those patterns, though I need to do a better job of following through by allowing students to test those conjectures until they find counterexamples or have worked their way to a convincing argument for why their conjecture is true.

To get at a true answer to Irrational Cube’s question, though, we need not just to better motivate proofs and give students a few authentic experiences to prove things, but to overhaul our math education system. We need these experiences — noticing patterns, making conjectures, struggling with them mostly unaided by the teacher, researching appropriate related material, crafting a series of flawed-but-each-one-better-than-the-last arguments for why the conjecture could be true, creating a good logical argument and then tweaking it to be airtight — to be the essence of mathematics learning from a young age. There should be less emphasis on proof in geometry (so that there is room for other great geometric topics) but more emphasis on proof in algebra, so that proof is not just an isolated island in a sea of algebraic solving-for-x. A great take on how we might make these radical changes is given by Paul Lockhart in his A Mathematician’s Lament, worth the buy in book form, but a shorter essay of which is also available for free here.

But, since in the short term anyway, an overhaul of math ed in the USA is not realistic, what steps can we take (as math teachers) to help students understand and value the role of mathematical proof?

Project-Based Geometry Outline

As a complement to my Geometry Skills List, here is my planned outline of Geometry projects. As I describe in a previous post, I’ve increased the project-based component of my math classes a little bit each year, and this year my goal is to have a sequence of overarching projects driving the course content. Direct instruction to provide background knowledge for the projects will fit in as needed, but not be the predominant mode of learning.

• Quarter 1: Patterns & Motion
• Week 1: Review of measurement skills, then 1-day (90-minute) Volume Project wherein students measure, convert, then find volume and surface area of a classroom
• Weeks 2-4: Tiling Project drives learning of regular shapes, transformations, & symmetry
• Weeks 5-6: Similarity, dilation, proportion are discussed, culminating in application mini-projects on creating scale drawings and indirect measurement
• Weeks 6-7: Fractal Project drives understanding of self-similarity, iteration, and fractal dimension, while reinforcing ideas of perimeter, area, and volume
• Weeks 8-9: Students use nets, toothpicks, gumdrops, straws, pipe cleaners to build polyhedra, then do the Euler Characteristic Project, plus we prove why there are only five platonic solids

• Quarter 2: Construction & Measurement
• Weeks 9-10: We discuss understanding measurement at a deeper level, converting between units of measure (dimensional analysis), choosing appropriate units, and measurement in the coordinate plane
• Weeks 10-11: Triangle Construction & Proof Project, where students investigate which pieces of information are sufficient to determine/construct a triangle; this ties directly into proofs of triangle congruence and other triangle-related proofs
• Week 12: Pythagorean Proof Project – After learning about and applying the Pythagorean Theorem, students research varying proofs and write a research paper detailing one particular proof
• Week 13: Sine Exploration Project connects with earlier discussion of similarity and proportion, leading  to an understanding of the three trigonometric ratios and their application
• Week 14: Students use triangles and triangulation to break harder problems down into easier problems (relating to area, angles, and art museum security)
• Weeks 14-16: Circle Construction Project deals with multiple ways of constructing circles and their measurable parts – students construct circles with compass going through 1, 2, or 3 points (connecting back to theorems about isosceles triangles and perpendicular bisectors), then construct, measure, and calculate lengths/areas for parts within a large circle
• Week 17: Return to measurement of volume and surface area; students empirically verify the relationship between prisms and pyramids (or cylinders/cones) with the same base
• Week 18: Students summarize learning in a culminating Portfolio

Hope this sounds interesting to you! I’m looking forward to creating some new projects and modifying existing ones to encompass more and better learning.

Geometry Skills List

Hi all. I am three days into my new year. In Geometry this year, I decided to combine a version of standards-based grading (SBG) with project-based learning (PBL – love those acronyms!).

I shared my syllabus with the class the first day, both as a detail-oriented hard copy and as a more thematic zooming ‘Prezi’ presentation. Now, this will be a very non-traditional Geometry class, idiosyncratic in many ways to me and my mathematical/engineering interests (while still aligning with Maryland state standards). The first half of the course will be focused on patterns and motion, with units on symmetry+tessellations, fractals, and polyhedral patterns; following a one-week fast-paced review of measurement. The second half of the course will look in more detail at measurement and also construction: of triangles, of other polygons built via triangles, of circles, and of some familiar 3D shapes.

But even though some topics are based on my individual passions (although I had considered it before, I was inspired/motivated to actually do a fractals unit after hearing Bob Devaney speak at MathFest this summer), I believe in some ways this represents a truer picture of the real field of geometry than a curriculum driven by axiom/theorem/2-column-proofs where some of the things that go in the reason column are complicated names for something completely obvious (think ‘transitive property of equality’) while others are intuitive leaps that most high school students would never come up with. Some higher-level study of geometries focuses on symmetry and the invariants of a space (e.g. which transformations are isometries and leave an object/shape fundamentally unchanged, and which are not). I took a math course in college that focused on tiling theory (think your bathroom floor tiles or M.C.Escher’s tessellations). Fractals are a new and currently expanding area of study, unlike most of a secondary geometry course that is over 2000 years old. I’m hoping our unit on fractals will change the misperception that all math is static, unchanging, and always has an already-known-by-the-teacher right answer.

Some of my sequence’s focus on measurement and construction is influenced by my recent foray into learning and teaching engineering, while some of it is based on envisioning what real-life skills my students can learn via the course. Additionally, I admit that one personal goal in teaching this course is to convince students not to hate math, so this has also been a factor in deciding how I will teach geometry on this, my fourth go-round.

All of that is a roundabout justification of some of my choices of topics. Grading-wise, I decided to go 50/50 skills quizzes and projects. I am a big proponent of project-based learning, and I’m making some changes this year so that the geometry class will become more project-driven and less of, every week, oh here’s a project that sort of relates to what you’re learning. I believe that projects can assess applications of math, connections to other subjects, and a deeper contextual synthesis of multiple math understandings that is not reflected in discrete standards-based skills assessments.  Perhaps one of the old hands at SBG can tell me how to refine standards based grading to get at these aspects.  But since I don’t know how to incorporate this sort of learning into SBG, for this course I will weight the projects 50% and SBG skills assessments as the other 50%.

Anyway, here’s my Geometry Skills List. I would love feedback as to any topics I excluded that you consider essential to a students’ understanding of geometry, to their knowledge of the world, or to their grasp of future mathematics. I’d also appreciate hearing if any of the skills I listed are less important, or if any should be broken up into multiple skills or combined into a single skill.

1. Student will measure lengths to the nearest 1/8 in or 0.1cm.
2. Students will measure and draw angles accurate to the nearest degree.
3. Students will classify angles by type.
4. Students will identify the number of dimensions an object has.
5. Students will use formulas to compute areas of various shapes.
6. Students will compute perimeters of various shapes.
7. Students will use formulas to compute volumes of various shapes.
8. Students will reflect, rotate, and translate shapes with a high degree of accuracy.
9. Students will identify reflectional, rotational, and translational symmetry.
10. Students will classify plane tilings by the shapes around a vertex, and by symmetries.
11. Students will use proportions to determine unknown sides in similar shapes.
12. Students will demonstrate an understanding of fractals and self-similarity.
13. Students will measure perimeter, area, and volume of fractals at various iterations.
14. Students will count the edges, vertices, and faces of polyhedra.
15. Students will use Euler’s Formula to predict aspects of unfamiliar polyhedra.
16. Student will determine an appropriate unit of measurement given a situation.
17. Students will convert between different units of measure by dimensional analysis.
18. Students will use distance and midpoint formulas for lengths in the coordinate plane.
19. Students will identify types of triangles, by angle and by side lengths.
20. Students will construct triangles given side and/or angle measures.
21. Students will use the congruence theorems to prove triangles congruent.
22. Students will apply the Pythagorean Theorem to find unknown sides in right triangles.
23. Students will use trigonometry to find unknown sides and angles in right triangles.
24. Students will construct circles passing through 1, 2, or 3 points.
25. Students will identify parts of a circle.
26. Students will calculate lengths and areas of a circle and parts thereof.
27. Students will use formulas to compute surface areas of various shapes.
28. Students will use formulas to compute volumes of various shapes.

I look forward to discussing these with you all. By the way, there’s still time to influence my decision and change the list–any but the first few that are on tomorrow’s quiz :^)

Project Based Learning in Math

Next to the catapult-building session, the best session at the National Academy Foundation’s July conference was Leslie Texas’s “Project-Based Learning For Math.”

The Presentation

We began the session by listing together some of the challenges or barriers to using project-based learning in math. The main two challenges seemed to be time (how to fit such projects into an already-overcrowded curriculum) and the test (mathematical discovery and real-world applications will not be tested, so they get bumped in favor of tested material). To address these, Ms. Texas advised viewing a project in a different way: not as something to be added in at the end where students apply a mathematical tool after it has been taught, but instead as an overarching framework that bridges many standards/concepts and drives students to learn those concepts. Students can be introduced to the project idea at the start of a unit, then work on it in pieces over the length of the unit, and the project will create the need (or at least motivation) for the relevant mathematics to be learned/taught.

The most inspiring part of the presentation was when Ms. Texas described a project she did with her students in a combined math & science class. She had students design and build a model of a bridge, which incorporates the physics of forces, and the math of similar triangles, trigonometry, and vectors. But most important was her idea by which she created an audience and a purpose for the project. She had read an article in her newspaper about plans to build a new bridge across the Ohio River near her town. And the local Department of Transportation was soliciting proposals from engineering firms in the area. So she called up the DOT and asked them if her students could be part of the proposal process. After some discussion and convincing, they agreed, and she asked if they could send an official request for proposals to her class. So then, at the start of the unit, she was able to show that letter to the class, and pass it around to convince the students it was real and they were really being asked to submit designs! Students created designs to the specifications of the real project, as well as slightly-less-accurate balsa-wood or spaghetti bridge models. They were asking Ms. Texas to teach them about the math and science of forces, so they could have a better design. In the middle of the project, Ms. Texas brought in engineer volunteers to look at the progress and give real critique/feedback. At the end, the students had a final product (report/design brief) of high quality and had learned lots of math and science.

In closing, she provided us with a list of resources that can be used for math projects, of which the following were interesting enough for me to write down:

• Engaging algebra videos by The Math Dude (site)
• Math raps at educationalrap.com (their math page, listen to a sample of “Circumference (It Just Makes Sense)”)
• The National Math Trail, where students map out a path, including pictures and worked solutions, of math problems in their local community (site, Kay Toliver explaining her use of the Math Trail)
• West Point Bridge Designer, a free software where students design and load test bridges while striving for cost efficiency (download page)
• Eeva Reeder’s geometry project where students design a school of the future (overview, project details, article by ER on applied learning)

My Reflection

Over the past four years of teaching math courses, I have moved to include more (and hopefully better) projects each year. This is partly influenced by my transition to teaching half-engineering, where the curriculum is almost entirely projects (with direct instruction as needed to aid the projects). But it is also due to my philosophy of teaching: I believe students learn better when they discover the ideas for themselves and when they engage hands-on in doing of mathematics. I have posted most of my Geometry and Algebra 2 with Trigonometry projects online.

What I would like to do this year is move away from including lots of projects interspersed within my math classes, to having the projects drive the math. I need to sit down and pick out maybe four-six major projects (either extensions of mine or from other sources) that have enough heft, and bring together enough concepts,  to drive the curriculum. In addition to choosing them, I need to think about how I can work to make them have a real purpose for the students and a real audience beyond the teacher, whenever possible.

Catapult Battleship, or Adults Like Having Fun Too!

Our school is working toward becoming an Academy of Engineering site, for which reason four of us from that academy attended the National Academy Foundation (NAF) Conference. The best session at the NAF conference that I attended was one on integrated interdisciplinary units that bring together both the PLTW Principles of Engineering (POE) course and other core academic subjects, put on by Pier Sun Ho of ConnectEd.

Half of what intrigued me was how to connect the POE curriculum to other subjects. For example, at the time the POE teacher was teaching about ballistic motion, the algebra or geometry teacher could be teaching about quadratics or trigonometry; the history teacher could be discussing World War II bombing of London and Dresden; the English teacher could be teaching argumentative skills needed in a debate; and the physics teacher could be teaching about trajectories. Not that all these subjects line up the same year in Maryland, but just the idea of weaving together so tight a connection among subjects was exciting! The ConnectEd folks even have provided a curriculum that tie these subjects together, available to all NAF schools via the password-protected myNAF website.

The other half of what made that session the most interesting is that we (teachers, guidance counselors, administrators, and business partners alike) got to work as a team to build catapults, then compete against each other in a game of “You Sunk My Battleship.”

A team's catapult, mounted on a 3' x 1' board

Details are as follows: Each team was given 15 notched popsicle (craft) sticks, glue, masking tape, 2 binder clips, 4 rubber bands, and a protractor, along with a 2″ square of cardboard with which to build a cup to hold the ping-pong ball. Then, after 20 minutes of design/building time, teams competed against each other by mounting their catapults in a fixed position to 3 ft by 1ft pieces of cardboard (their ‘ships’). The teams lined their ships up along the carpet (or tiled floor), then proceeded to do battle. On each turn, a team could move three spaces in one direction, rotate 90 degrees, and/or fire the catapult, in any order.

You Sunk My Battleship

Besides learning how to better collaborate with other subject teachers while I teach the POE course, I also realized during this session that even adults enjoy having fun and hands-on activities too! Similarly, one of the best parts of a conference I attended over a year ago in Atlanta was when teams of us got to build a toothpick-and-jelly-bean tower, with the goal of using the fewest toothpicks to successfully build a tower of four stories that could support the weight of a baseball for at least thirty seconds. Related to this idea, to help improve my classes I have worked at including more short mini-projects in POE, to complement and motivate the PLTW curriculum, as well as helping run the annual STEM Competitions.

But perhaps we could use this idea for adults too? Adults in the education field have enjoyed the toothpick tower and catapult at professional development sessions. In the past years, we in the engineering department have held a parent/family orientation session to let families know what the PLTW engineering pathway is, and what their child will be involved in over the nest several years. Perhaps we can expand that orientation session to include a hands-on engineering mini-project, so that families can experience a bit of the engineering design process that their kids learn about. If they are like us, they will not only learn about the engineering pathway but also have a lot of fun!

Engineering Overview

In my school, I teach as a dual member of the math department and the Project Lead the Way engineering department.  Our students in engineering for the most part have chosen engineering as their pathway (at the end of their ninth grade year), and take the courses in the following sequence:

1. Principles of Engineering (POE): fall, grade 10
2. Introduction to Engineering Design (IED): spring, grade 10
3. Computer Integrated Manufacturing (CIM): fall, grade 11
4. Digital Electronics (DE): spring, grade 11
5. Engineering Design and Development (EDD): spring, grade 12

Our school is unusual in offering POE as the first course. Although Project Lead the Way (PLTW) does not require a specific course sequence, most schools begin with IED.

Why do we do it differently?

Principles of Engineering, as can be gleaned from the course title, is a survey course, where students learn what engineering is all about.  They learn about some of the types of engineering (civil, mechanical, materials, electronics/computer), by completing a major project in each area.  They learn about the engineering profession.  They learn about the engineering process (briefly: design, build, test, then refine).  They get a sample of many aspects of engineering that will be studied in more detail in later courses.

Introduction to Engineering Design is not a survey course.  Its focus is more narrowly on design, which leaves out the key steps of building, testing, and re-designing (based on the test’s results).  Compared to POE, it does not give students as good a picture of what they can expect from our engineering pathway.  The argument for IED first is that it does build students’ requisite engineering skills:  skills in the design process, and skills in use of the Autodesk Inventor software.  These skills are used in many of the later courses, and by actual engineers.

Math Math Math!

I feel strongly that POE is a better introductory course, though I would be thrilled to hear arguments for IED (or even for other courses) in the comments.  However, there are some drawbacks to having POE first, and there are a few new ideas we are trying out in an attempt to overcome those drawbacks:

POE is very math-intensive.  Most of our tenth graders are taking Geometry concurrent with POE, having passed Algebra I.  This semester, I am teaching a class of ninth graders, many of whom are still in the midst of learning Algebra I.  The mathematics in POE ranges from right-triangle trigonometry (usually not seen until Geometry class), to work with vectors (usually not seen anywhere in high school math, maybe in Algebra II or Precalculus), to variable substitution in formulas (Algebra I), to measurement (elementary school, though many students still struggle with reading a ruler), to work with parabolas and quadratics (Algebra II).  What this means is that the POE teacher is often teaching these math skills before the student learns them in math class.  Additionally, the use of such a multitude of math skills, often in combination¹, requires a mathematical sophistication that most 9th-10th graders lack.

All in all, this sounds like an argument for leaving POE until the 12th grade!  That way, students will have had Geometry and Algebra II, will have built up that mathematical sophistication, and may have taken /  be taking a physics course.  But this doesn’t seem realistic either, since POE is by nature an introductory survey course.  It has hands-on projects that give students experience with and insight into the varied fields of engineering.

Here are two strategies I am trying out this year with my ninth graders, in an attempt to address the mathematics dilemma:

• ‘Math Boot Camp’ at the start of the course
• Friday Mini-Projects

Math Boot Camp

Since even tenth graders struggle with the math content of POE, I figured my ninth graders wouldn’t have a chance of surviving the symbolic and calculational onslaught unless I prepared them well at the very get-go. I created a five-day, five-topic overview of some of the math needed in the course.  Day 1: a solving equations [Algebra1-style] review [in the form of a math puzzle].  Day 2: an engineering formulas row game [moreinfo on row games].  Day 3 I intended as a lesson on dimensional analysis, but I never developed the materials because the snow daysinterfered with my plans.  Day 4: an overview of measurement with rulers, protractors, and dial calipers.  Day 5: an introduction to trigonometry.

I definitely believe that devoting the first week of POE to this ‘Math Boot Camp’ has been helpful in preparing students for the math they are now facing in class.  Still, it is important for me to continue to reinforce the math throughout the semester, so students will retain the new mathematical knowledge and be able to apply that math in engineering contexts with ease.

Friday Mini-Projects

The other issue with having such difficult math concepts in an introductory class is that students can get turned off easily.  If they are bored, or don’t understand (or, more likely: both), they will tune out from a teacher’s explanation, and withdraw their investment in the class.  To avoid this, I try to be very animated to keep students’ attention when I present a complicated math topic (like truss calculations) at the front of the room, and involve as many students in the process by asking little questions out to the audience.  I also try to mix it up, so students are actually building truss bridges of balsa-wood on the same days they do truss analyses.  They don’t spend all ninety minutes on the mathematics, maybe 45 on truss calculations and 45 on bridge building.

This semester, with help from my colleague Ms. Ball, I am adding a new item: the Friday Mini-Projects.  Now, POE’s curriculum (developed by PLTW) is already super-crowded, and we lost several days to the snow, few of which will be replaced.  So it may sound crazy to be adding, rather than subtracting!  But there’s a method to my madness.  These Friday mini-projects are 1-day engineering projects that will sustain and excite student interest while connecting to key points in the standard POE curriculum.  For example, the Toothpick Tower can be designed and built in thirty minutes, tested in ten more.  It connects to the unit on structures, statics, and trusses, allowing students to see (some for the first time) how a triangle is really stronger than a square.  The mini-projects also connect to and really reinforce the engineering process (Design, Build, Test, Refine; or the more detailed 10-step version) because students follow that process, via a guided report they complete for every mini-project.

The Friday Mini-Projects are designed to motivate students via a quick engineering problem-solving competition.  They can really help if students start to get bogged down in the advanced math and physics concepts involved in the course, by drawing the students’ interest back in, while still advancing the students’ engineering skills and the POE curriculum.  They can all be completed (and sometimes the reports too) in one ninety-minute block period.  I have used one or two of these in my teaching before, but this is the first time I am implementing them systematically:  nearly every Friday, with their own weighted category in a student’s overall grade.  So far, they are going well (and I thank Ms. Ball for the ideas and collaboration!).

My goal, with the Friday Mini-Projects, with the Math Boot Camp, and even with keeping Principles of Engineering as the initial course in our sequence, is to spark student interest in engineering, while maintaining the course’s and program’s depth & rigor. I will keep you posted on my success or failure toward reaching that goal, as time goes by.

1: For example, analyzing truss diagrams involves vector arithmetic, trigonometry, proportions, summations of signed values, and the Pythagorean Theorem.

Snow Daze

(Trying to get this blog thing going again …)

As you probably know, Baltimore was hit last week by a pernicious pair of blizzards, dumping a total of 45 inches of snow on us, something this city has not seen in decades if not centuries.  Although national news coverage focused more on the ‘snowpocalypse’ in Washington, DC, Baltimore and its surrounding counties were more heavily hit than our nation’s capital.  This brought us briefly to the top of the list as #1 snowiest city for the 2009-2010 season, ahead of even Syracuse and Buffalo!  Syracuse has now edged us back out for the top spot.

We lost all five days of school to the storms last week.  This Tuesday (16th Feb), after the Presidents’ Day holiday, schools were still closed due to icy and unplowed roads, as well as unsafe conditions along sidewalks and bus stops.  Even though school re-opened on Wednesday, with two-hour delays scheduled through the rest of this week, I think it’s clear that Baltimore City was not prepared to handle this amount of snow.  Side streets remain plowed only enough to allow one lane of traffic at a time on a two-way street.  Major thoroughfares that are  usually three lanes in each direction only have room for one lane of traffic in each direction, due to piles of snow just pushed off to the side.  Sidewalks are still not shoveled, leaving kids to walk in the street to get to school or their bus stop (and me to walk in the street to get to my car).  Some pictures and commentary on the road/sidewalk situation are here.

As a teacher, I’m concerned about how this will affect what I’m able to teach.  I’ve already thrown out the question on Twitter about how best to compress my curriculum in Algebra II with Trigonometry.  I welcome any further feedback on this:  which topics I could cut or skim by with just a pass, versus which topics are core ideas that need thorough investigation.

I’ve talked to my engineering teacher colleagues about how to condense some of the Principles of Engineering curriculum while still conveying the core ideas and experience of engineering in this survey/introductory course.  What makes this one even more difficult is that I need to prepare my POE students for a standardized, end-of-course exam, while I create my own final exam for my A2T students.

If we were to make up all the (nine) snow days we have missed so far this school year, I would be less concerned about taking a hatchet to my curricula, because I would only have to account for the six half-days missed this year (from snow delays and early dismissals).  But there are several indications that most of the instructional time missed due to snow will not be made up:

1) Last year, the 3-4 snow days we had were added to the end of the school year in June, after final exams were over and grades were turned in.  There are always a few such days there anyway, nominally “regular school days”, but where student attendance drops down to 10% since courses and exams are over.  These days are really (in everything but name) days for teachers to get ready for summer by packing up their rooms and completing several end-of-year record-keeping rituals.  Since last year’s snow make-up days were added on at the end, without final exams being postponed, this led to an exorbitant seven(!) post-exam days with hardly any students.  If we add our nine and counting snow days this year without delaying final exams, this will be wasted time.

2) Our state superintendent has already declared that she will approach the state school board seeking a waiver of the requirement that students attend school for 180 days.

So, here’s hoping that at least some of the days will be regained in an instructionally meaningful way!  In the mean time, please contribute your thoughts here on what to cut and how to streamline my algebra and engineering curricula!