Day 59: Discussion

Wanted to spend the hour going over U4 WS1, but so few students have bought in to the idea of struggle and work outside of the classroom, that we had to spend the hour simply giving class time for students to do the work. 

Again, the prevailing mindset is that if something looks new/different, that must mean that they (the students) are not capable of completing the problem. They absolutely refuse to even attempt to forge connections from notes & previous examples in order to tackle a new challenge. And of course, they then become very upset with me that I won't simply show them how to do it.

Couple of things I noticed from student questions: 

• Some students are confusing area with distance. When given a picture of a triangle drawn on a grid and asked to determine the length of a side, they'll count half-boxes at the edges. Don't think I've ever seen that one before (but I've never relied so heavily on examples from Geometer's Sketchpad either). I might need to pay better attention to vocab in the future ('spacings' instead of 'boxes'). 
• The idea of a model is so foreign to most students that they're unsure of how to transfer it to new situations. We created the model for diagonals & side length of squares in Unit 3 for the express purpose of applying said model in this unit with isosceles right triangles. Students have the model written down in their notes and remember using it, but are unsure of how to apply it to a situation that isn't 100% exactly the same (since these are triangles, not squares). 
• I really hate how much calculators are foisted upon students these days. They have very little of what I call "number sense" because the magic little doohickey always solves the problem for them. Case in point: the square root of 2. It's almost as though students see the radical sign as some sort of variable that MUST be dealt with. I've been trying to hammer home the idea of precise vs. approximate (as a result of rounding) and why it's totally OK (and even preferred) to leave answers in radical form. My non-honors class literally lost their stuff when I tried to explain the difference.

I'm guessing Unit 4 will take up most of my time leading up the holiday break, but I don't want to start a new unit before 2 weeks off. We only have 9 school days after the break before the end of the semester, so I'll need to work out if/how much of unit 5 we'll get to before the final exam. 

Day 58: Online Practice

Spent the day in the computer lab with the students working with the Carnegie Learning software. Did this mostly because my 5th hour was a day behind the other classes, so I gave them the class period to catch up while the others worked online. 

I created a custom sequence of the Geometry units in Carnegie to better match the sequence I had written for my class. The only problem is that because I don't "force" the students to complete Carnegie by given deadlines, they don't work very hard. And now, 3 months into the school year, most students are still in the first unit of Carnegie, when the class material is aligned to the fifth unit of Carnegie. When asked, I told one student that Carnegie is just practice. If you choose not to practice, you're choosing to not be at your best.

We're going to start having skills review quizzes next week, so maybe the poor pacing of the online practice will actually come in handy. 

Day 57: Area & Perimeter

I love it when a plan comes together. 

We started class with a recap of the challenge from yesterday. I recorded the legs and estimated areas from a selection of students, and then solicited ideas for how to relate the three quantities. Because of the estimation with partial box counting, very few triangles worked perfectly, but they worked. 

Of course, most students were stumped after trying addition, seeing that it didn't work, and not knowing what to do next. A couple remembered the correct equation before we even got to this point, but a select few actually stumbled across the connection and didn't make the A=1/2 b*h connection until we were done. 

It's fun getting to see that light bulb go off; even more so when it's something "simple" that they saw in middle school and didn't recognize. It's as if most students don't even realize that there IS a back story to where the content comes from. They're so used to being spoon fed a bunch of stuff that they're expected to memorize and regurgitate, that they never stopped to think about what's "behind the curtain" so to speak. 

And that's kinda my whole point in doing this, so I'm calling today a win. 

Day 56: Unit 4 (Right Triangles)

Ok, it's time to actually start covering material that might be new to students. I started by stressing the purpose of my curriculum because many of the students have noticed that I'm doing everything differently than the other geometry teachers in school. We're doing things differently so that everything is built from stuff we've learned. Unit 3 was all about squares, so we're going to start Unit 4 from there and go forward.

Major point to stress in the "review" is the diagonal to side length ratio. Looking over the U3 Assessment, many students still count diagonal spaces on a graph in the exact same fashion as horizontal & vertical spacings. 

So with a diagonal drawn in a square, we very obviously have two triangles. Coming from a square, we know the sides are equal and there are 90 degree angles in the "corners." Intro the vocab that 2 equal sides --> isosceles, so we can call this triangle an "Isosceles Right Triangle" (IRT). 

Also coming from the unit on squares, we know that the diagonal acts as an angle bisector. Half of 90 is 45 degrees, so each acute angle in the IRT must be 45 deg. 90+45+45 = 180 which conforms with expectations since the four 90 deg angles in the square summed to 360, and we cut the square in half. 

I'm trying to keep everything very explicit as we build the unit from the ground up. I make a point to stress that we only know the angles of an IRT sum to 180 because we only knew the 4 angles of a square sum to 360. We can branch out later and verify that the idea holds true for all triangles. 

I then tasked the class with a challenge. Draw a right triangle that is NOT isosceles on graph paper, keeping the sides to integral lengths. Determine the area by counting boxes (remembering that we defined area in U3 with squares) and try to find a connection between the lengths of the legs of the right triangle and the area. 

I really thought kids were going to just start with the assumption that A = 1/2 b*h and claim they were done, but they actually didn't see the connection, presumably because I never used the terms 'base' and 'height.'

We'll recap the challenge at the start of class tomorrow and start work on U4 WS1: Area & Perimeter of Right Triangles. 

Day 55: Standardized Testing

WAAAAY back on Day 3, I avoided posting a long rant about the standardized testing that took up an entire class period. I'm going to try and avoid the full rant, but rather offer an explanation of what's going on. 

I'm required to give a Quarterly Common Assessment to the core classes I teach (physics and geometry). This means I give the exact same test at least 5 times throughout the year to my classes so that the school can track growth. 

For physics, I use the Force Concept Inventory which is far better than any assessment I could write. But for geometry, I volunteered to create a diagnostic / common assessment that all the math teachers could use. Right now it's a 45 question test with 15 questions covering pre-algebra, 15 covering algebra, and 15 covering geometry. The original idea was to give a diagnostic test to students who enter our school at random times throughout the year so that we could place them in the appropriate math class. I co-opted the idea for the common assessment for "simplicity."

To recap: I gave a 30 question test (no geometry) in the first week of school. Today I gave those exact same 30 questions with an additional 15 tacked on. I will give that same 45 question test again at the start of the 2nd semester (don't want to waste time before finals), again at the start of the 3rd marking period (early April I think) and again before school lets out in June. The exact same test. 

There's a debate here about the merits of "growth" when measured like this, and I think my opinion shines through with the emphasis I added. 5 class days lost to (IMHO) invalid data. 

Day 54: Assessment

Unit 3 Assessment. 

My prediction: students will struggle with the length of diagonals in squares, creating images with symmetry, and solving for properties of squares when NOT initially given the side length. 

My goal for the Thanksgiving break is to create the U3 Reassessments and the structure of Unit 4 so I can be ready for the rest of the year leading up to the X-mas break. 

Day 53: Review

Task #1: finish going over U3 WS4 from last week.

Task #2: Give students time to create a note sheet for use on the Unit 3 Assessment tomorrow. 

With Task #1, I tried to make the deeper connections that I had hoped for when we started this unit. Through some guided discovery, the class was able to see that lines of symmetry have to be angle bisectors on both ends, perpendicular bisectors on both ends, or one of each on each end. That's something at least. 

With Task #2, I stress that even though students are allowed note sheets, less than half of students actually use them on the tests. If there's a greater indicator of what students are willing to do to be successful, I haven't found it yet. Of course, some students who don't use note sheets are happy with grades in the C range, which is another issue altogether. 

I'm ok with note sheets as I'm confident that I can write a test that requires a deep understanding of the content to be successful. Were I giving a simple true/false (or even multiple guess), then yeah, note sheets might be too much of an advantage. But that's why students hate my tests (and in some cases, by extension, me); because they can't be aced just by memorizing a bunch of facts and properties. Heck, they generally can't receive good grades by even memorizing a procedure (like, always add two numbers and divide by 2 to get midpoints). Of course, this is exactly my goal, so I'm ok with it. 

Day 52: Origami

Easy day. Work through some origami creations while identifying types of symmetry in things that aren't everyday common shapes like squares. 

If you'd like the link to the website I used (downloadable instructions and animations) as well as any worksheets I've created, check out www.mrfuller.net and search the past calendar under Geometry. 

Day 51: Discussion

Very unproductive day. Goal was to work through a representative sample of the worksheet problems so that we would see all the connections there were to see. Unfortunately, so many students had completed so little of the assignment, that we weren't ready to go over it. By the end of the class, we'd only managed to get through 2 or 3 problems. 

I'm still very happy with my progress under a new curriculum. I'm very excited to see how the new sequence unfolds based on how structured the content has been thus far. However, I'm more than a little concerned that as the course content builds, students that haven't been working to their full potential are going to find themselves simply unable to participate in class (if you can't identify the slope of a line and you can't solve for the midpoint of a segment, how much luck are you going to have discovering general properties of parallelograms?). 

I think the biggest lesson I'm learning this year is that the method of delivery and the sequence of the content are not the deciding factors in whether or not a student is successful. Do whatever you want - "flip" the classroom, lecture, model, whatever. If the students have no appreciation for the importance of education and are not willing to struggle to reach a desired goal, no fancy new approach will work. 

Sorry, I try to avoid being negative, but it's the start of the 2nd marking period and I've been reflective this week. 

Day 50: Symmetry in Practice

Very loosely structured day. Spent the first half just trying to firm up our ideas about symmetry from the intro day yesterday. Settled on a basic definition that symmetry results from any action that creates a result indistinguishable from the original image. I tried to word it like this to avoid pigeon holing us into thinking that symmetry only dealt with folds and lines, because rotations are just as important. 

After the recap, I had students work on Unit 3 Worksheet #4. Some students work, but many did not. Some would work through the first 4 problems, definitely the easiest problems they'll ever see. Then they'd quit, either thinking that the content is so easy that they don't need to bother doing any more, or being so completely stymied by the transfer of skills to connect with midpoint and angle bisectors that they lose all hope. 

I make a point of talking about each of these pitfalls and stressing the need to work through struggle and try to practice every style of problem to make sure that you understand all aspects of the content, but my pleas often fall on deaf ears. 

Day 49: Symmetry

I actually spent a fair amount of time prepping myself for this unit because I had high hopes of delving very deeply into symmetry. I had never really thought about the underpinnings of symmetry and how you can discover properties that can be proven geometrically. I found some lessons online that got into perpendicular and angle bisectors and read them through enough times that I felt confident. I planned out the week to build to cementing these ideas and prepping for the Unit 3 test before Thanksgiving. And then, I went through the intro day. 

Nothing bad happened during the intro day, in fact, I would probably label it a success. I probed for students preconceptions of symmetry, jotted down the class ideas and worked through some basic examples of line and rotational symmetry. But at the foundation, I realized that what I had hoped to accomplish was way beyond what most of my students were capable of getting through, at least on the order of a few days. 

So, I decided to re-plan my week to focus on making connections between symmetry and everything we'd talked about in Unit 3 to date, bisectors and midpoint to name a few ideas. Let's hope that works. 

Day 48: Recap of Unit 3 to date

Between the students who never take notes (and somehow still find ways to get upset with me when I won't answer straightforward questions such as "what's an angle bisector?") and the students who miss class (and never both to check the website or ask a friend for the material that they missed), I used today to walk through the entirety of Unit 3 to date, writing out 5 pages of review notes on the tablet (so the file could be uploaded to the class website to never be heard from again). 

Unit 3: Squares (so far)

• Definition
• Properties (Area, Perimeter)
• Diagonals (relation to side length, perpendicular bisectors, angle bisectors)
• Midpoint formula
• Using bisectors outside of squares

And now we're ready for symmetry. 

Day 47: Online Practice

Back to the computer lab for more practice online with the Carnegie Learning Software. 

I stress to the students that most are stuck in Unit 1 online (linear equations) and that I set up the software to run alongside the class which is in the equivalent of Unit 5 online. Of course, they want to be skipped ahead to catch up, but I refuse pointing out that they're behind because they don't use class time efficiently and because they're not practicing at home, which is one of the major reasons we have the online tutoring software. 

Surprisingly, my regular geometry class was actually the best behaved and most on-task in the lab. I might readjust how I spend time with that class and get them in the lab more often. 

Day 46: Discussion of Angle Bisectors

We spent the day going over U3 WS3 on Angle Bisectors. I have about 15 minutes of class for students to work, and then I led Q & A for no more than the setups of some of the problems. We did not do anything like a whiteboard session due to time constraints. 

I was very surprised at how little transfer students are willing to even attempt. They can recite the definition of an angle bisector, but cannot use that information to set up an problem involving algebraic expressions for 2 angles and solve for the unknown. They claim "I don't get it," but as I walk through the questioning techniques with them ("what is an angle bisector,?" "what does that mean for the angles,?" "how would that be set up in an equation?") it's clear that they do "get it," at least at a very basic level, they're just either completely incapable of, or simply refuse to, attempt to apply things they know to new situations. 

Day 45: Geometric Constructions

Every year I subject myself to a lesson along these lines, every year I get the same result, and every year I keep coming back for more. Didn't Einstein have something to say about that?

Anyway, I finally decided to bust out the compasses, but only for my honors classes. I really think the compass is an amazing piece of technology and truly underscores the idea of congruence without equivalence, but those ideas are incredibly abstract and very difficult for my students to grasp, so the lesson almost never lands the way I hope it would. 

I started by showing the different varieties of compass and emphasizing that they all do the same thing; create a set of points that are all equidistant from a common point. Using just a compass and a straightedge, the Greeks were able to discover most of classical geometry in the absence of a decimal numbering system. Pretty awesome, right? Well, not to 15 year olds anyway. 

I demo the basic construction of copying a line segment using a document camera. At this point, I haven't given out the compasses, because all heck will break loose when I do (I've learned at least that much in my years trying to do this). I have a pre-printed packet of 4 basic constructions that I hand out along with the compasses, and then we walk through creating a parallel line through a point while showing a flash animation on the screen. 

No joke, it actually worked a little this year. No idea why, but the results were more along the lines of what I hope for. What was lacking was the appreciation aspect of it. Kids don't really care about why a compass is useful or what we can do with it. Nobody wanted to explore the rest of the constructions at the end of the period or see what else they could do. 

I hope I can devote another class period to at least the angle bisector construction, but it's hard to justify spending so much time on a skill that isn't really going to enhance their understanding of the content. 

Day 44: Angle Bisectors

I spent a lot of time thinking about this lesson. I really wanted to stay true to my goal of making the content sequential and discoverable, but I wasn't sure how to arrive at the idea of an angle bisector without so much else in the course. I finally settled on a plan that I was happy with, but it was more convoluted than the students could handle and I'm not sure it will end up being worth the hassle. 

Here's what I did:

• Have students draw square ABCD that is at least 5 x 5 on graph paper. 
• Draw diagonal BD
• Place four points anywhere along BD and label them F,G,H, and I (in order)
• Measure the distance from each point to the opposite corners A & C (so there are 8 lengths total)
• Make conclusion

The goal here was to kind of back in to the notion of angle bisectors, but all this part of the activity does is conclude that points along the diagonal of a square are equidistant from the corners. How can you show that the angles are in fact equal? That's where I got stuck.

So I very briefly addressed the idea of triangle congruence using SSS. If the sides are all the same, then the triangles have to be the same, and if the triangles are the same, then the angles have to be the same. But in actuality, that only deals with corresponding angles - I'm pretty sure you'd need to dig deeper, into the Triangle Sum Theorem to prove that each angle was 45 deg and bisected from the corner of the square. 

No school tomorrow (Election Day), so I'll try to think of some way to tie it all together so we can close out Unit 3 with angle bisector construction (pray for me) and symmetry. 

Day 43: Discuss U3 WS2

Goal was to go over the worksheet assigned yesterday. Surprisingly, most students attempted at least the easiest problems (finding the midpoint from the graph and when given two endpoints). 

I've settled into somewhat of a groove in which I handle worksheet discussions different in the honors classes vs. the regular class. In honors, I have each group whiteboard ~3 problems and utilize the "whiteboard parade" method in which the kids have a chance to walk around the room and check the other boards and ask questions. There is almost too many kids and too many simplistic problems to justify the "one WB presentation at a time" method. 

For the regular class, I basically can't relinquish control of the class like that. I need to maintain order and that means that we actually avoid group work of that nature. Instead, I passed my tablet PC around the room and had randomly picked students solve problems on the tablet while it projected to the screen. They like using the tablet, but there's still the issue of what students choose to do while waiting for someone to write their solution down. 

Day 42: Midpoint (cont.)

Agenda item #1: U3 Quiz 1 on the basic properties of squares.

Agenda item #2: Recap the Midpoint discovery w/GSP from yesterday. 

Agenda item #3: Start work on U3 WS2 - Midpoint

As mentioned yesterday, it was incredibly surprising at how little math sense students have. They can visually point out the middle of a segment, and they appear to understand the concept of average, but are generally unable to combine the two ideas. My theory is that they have only experienced math as a set of seemingly unrelated procedures one follows to arrived at a previously known conclusion. 

I stressed the idea of the midpoint formula being a model that we've built that *always* works. It's not a procedure in the sense that "you should always take two numbers and add them, then divide by 2," because that procedure falls apart if you're solving for the other endpoint when given the midpoint. Instead, if you use the model as derived and substitute in the knowns and solve for the unknown, you can't go wrong. Judging by their quiz results, they bought into that methodology, but the algebra skills are still fairly weak.