Wednesday, October 31, 2012

Day 41: Midpoint Discovery

The original plan was to have students continue on with the same graph of their square that we've been working with for a week now and determine the relationship between the coordinates of the endpoints of the diagonals and the coordinates of the midpoint. Students being what they are (teenagers), so many of them have lost their graphs or made such a mess of them that to continue working with the same data would be a fruitless venture. 

Instead, I took the classes to the computer lab to do the same activity using Geometer's Sketchpad. As much as I love GSP, I don't love trying to teach students how to use it. It's an incredibly powerful program, which means there are a TON of little things you need to know how to deal with if you want to get consistent results. Sadly, even with the most direct instructions, my students see the program as an obstacle and will often quit at the first sign of struggle. 

In the end, most students got the construction created successfully, but struggled with the discovery aspect of the activity. Most of them really fight the idea of being creative in math, they are so accustomed to being spoon fed a procedure which must be learned and memorized to be regurgitated later, that the freedom to create something new is a foreign concept in this specific environment. 

Some students did arrive at the idea of "middle," but couldn't make the mathematical leap to the idea of 'average' (further reinforcing my suspicion that even students who are generally "successful" in math have little idea what they're actually doing).

I will adapt the instructions to include more scaffolding, specifically the suggestion that the x coordinates should be considered separate from the y coordinates. 

Tuesday, October 30, 2012

Day 40: Practice and Discussion

Students worked on U3 WS1: Properties of Squares. This was something of a catch-up day so that all the classes could get back on the same schedule. 

This was a worksheet that I created over the summer when I was still in the planning stages of this whole experiment and unfortunately, I did not look back over it between the development and the deployment stages of the unit. What that left me with was a worksheet that was essentially a rehash of the discovery activities we'd been working on for the past 4 days. Oh well. 

Monday, October 29, 2012

Day 39: Practice with Squares

Today was spent recapping the conclusions we made last week and giving students time to practice with U3 WS1 - Basic Properties of Squares. 

The only "new" content discussed today was the relationship between the length of a square's diagonal and its side length. 

Student minds always surprise me - when I asked the class to try determine the length of a diagonal by counting boxes (like they'd done with the sides), I'd expected them to be stymied. "But Mr. Fuller, those boxes (corner to corner) aren't the same as the horizontal/vertical ones" I imagined them saying. Nope - they'd declare an answer - 5 (for example) - which always happened to be the same as the side length for the particular square we were looking at. And then it hit me: Because it's a square, the diagonals have a slope of 1 (or -1) and they were simply counting the spaces on the grid that the diagonal passed through, just like we did for the sides. Huh. 

So let's run with it: Projecting at image of our graph onto the screen (using Geometer's Sketchpad, so it looks precise), I can stand at the screen with a meter stick and have a volunteer who's close by read the length of a side. "57 cm" they'd say. Ok, now I'll pivot the meter stick at the vertex until it aligns with the diagonal. VERY clear that the 57 cm of the meter stick that's exposed doesn't reach the far corner of the square. Conclusion: Diagonal lengths are longer than horizontal/vertical lengths. But what's the relationship?

At this point, my original idea (back in July when I was still optimistic) was to have students "discover" the scale factor of sqrt(2), but I was running short on time and we needed to get going. Instead I worked at the computer and took suggestions (using GSP) for what mathematical operation might connect side length to diagonal length. This is why GSP is awesome - I can perform a calculation and leave the result on the screen while I alter the square to see if the result changes. I couldn't ask for a better visual for testing a mathematical hypothesis. 

Anyway, we try adding the side length to the diagonal and see if the sum is a constant. Nope. Subtraction? Nada. Multiplication? Still nothing. Division? Jackpot. Out pops 1.41 as this unshakable constant. 

Here's where I finally broke down and became "lecturer" for a moment. I asked: "If that number had hypothetically come out as 3.14, what would you think?" Students immediately jumped on Pi. Awesome. So I explained that there are a handful of very famous numbers in math that you'll start to recognize as you spend more time with math. So I lead them to the idea of sqrt(2) and thought we were done. 

Nope. The number of students that have NO IDEA how to work with exponents and radicals in a 10th grade geometry (honors or regular - it made no difference) class was astounding. Worse yet - there was no shame on the part of the students for not remembering the concept. Instead, they got frustrated with me that I wouldn't simply tell them the answer, insisting instead that they should be able to figure it out or ask a neighbor for help. 

Still a solid day. 

Friday, October 26, 2012

Day 38: Discovering Diagonals

The goal seemed so simple: task the class with creating three hypotheses related to the diagonals of a square. Problem #1: Students ignored that bit about diagonals and just regurgitated previously known facts like the sides are congruent. Problem #2: Students don't make any distinction between an obvious observation and a hypothesis that will require investigation. 

The idea was that with each student having their own unique square, any hypothesis they created could immediately be checked, albeit informally, with other squares for validation. The three conclusions I was looking for were: pairs of diagonals are always the same length; the point of intersection cuts the diagonals into equal pieces (haven't defined the word midpoint yet); and that the diagonals are perpendicular. 

The first two are fairly easy for students to "discover" if they're willing to put forth the effort to measure the segments. The last one is often thought up, but students have no idea how to prove it. I tried to lead students to the connection between "90 degrees" and "perpendicular" and hope they make the connection back to Unit 1 and see slope as a method (these squares are on graphs for a reason). 

For each class I made the brief point about why we spent these 3 days the way we did - I could have simply told everyone those formulas and conclusions on Day 1, but in the long run they wouldn't have a deep understanding of what they were actually doing. Some kids bought my explanation, but others (generally the regular Geo class) would just complain "this is stupid" and "just tell us the answers already." 

I don't know how to reach kids that have such a combative attitude toward learning. All the literature I've read claims that if students can take ownership of their education, they'll change their attitude, but I have NEVER seen that work in practice. I still think this is a better way to teach, but I'm stymied about how to handle students who actively fight against the class methods. 

Thursday, October 25, 2012

Day 37: Graphing Data

Just because this is a math class, it doesn't mean that students shouldn't 1) be increasing their graphing skills and 2) learning how to analyze data to look for evidence that support a hypothesis. 

So we built on yesterday's activity and collected the distance around the outside and the boxes inside every student's square. While displayed on the screen, I tasked the class with graphing "boxes inside vs. distance outside." There was a brief review of the importance of scale and which axis is which, but most students did just fine. There were some gripes that they had to plot *gasp* 25 data points, but I stressed that we needed a lot to smooth out the relationship in case anyone made mistakes with their measurements (which happened more than it should have). 

In the end, I used a document camera to show the results of their hard work. The only major conclusion I was hoping they'd get to is that the relationship is NOT linear. They should have learned about parabolas and what general equation would create what we're seeing, but very few did. 

I then used Excel to create my own graph of the same data. I did this so that I could easily show different potential trendlines to see which looked the best. I also displayed the correlation coefficient, but only described it as "the higher this number, the better the fit." Students quickly saw that the polynomial trendline actually fits better than the linear (it helps to have a couple of students make BIG squares to make this obvious). Then I had Excel provide the quadratic that describes the curve, and explained that our 'y' variable was really "Area" and 'x' was "Perimeter." They obviously knew these words, but seemed to be OK with my intended misdirection. 

The end result is A=0.06(P^2) (the constant will vary based on the data, the three classes hovered around 0.06). 

I continued on with a deeper analysis that I stressed was just FYI - students would not be responsible for doing this on their own. Essentially, I approximated 0.06 to 0.0625 so that it could be written as 1/16. Including the constant with the P^2, I rewrote the equation as (1/4 P)^2 which combines with the idea that P = 4s to create A=s^2.

Looking back, the honors classes were able to follow and at least some appreciate the derivation. This was a complete cluster in my regular class and threw everything off track for a full day. They simply don't have the attention span or the appreciation for something that's not required, as those students often struggle with connecting enhancement with greater success. 

Wednesday, October 24, 2012

Day 36: Intro to Squares

Ok, time to actually start implementing the modeling part of the class. My main idea for this unit was to take everything we've already covered in measuring distances and understanding angles and start out with the simplest possible shape, a square, to help students get comfortable with the idea of discovering the content on their own. 

We started making sure that everyone knew the definition of a square. I know it seems simple, but I learned a long time ago to never assume students know anything they "should" know. Students had a lot of good (and scattered ideas), so to ensure consensus, I simply googled "definition of a square." Google is becoming so powerful that it didn't simply give me a link to the answer, it actually just answered the question. 

Then I took suggestions for the different ways we could measure the lengths of sides. I wanted students to understand that there are multiple representations of the same idea. Students suggested the typical meters, cm, feet, inches, etc. Some suggested simply counting boxes on the graph which was nice to see. Nobody suggested using some form of coordinate subtraction , so I led the class to that idea. At this point, I kept things horizontal/vertical, so we're not getting into the actual distance formula just yet, just the ideas behind the Ruler Postulate (I do NOT use that phrase in class). 

From here we transitioned into other things we could quantify from our pictures. I asked all students to count the total distance "walked" around the outside edge of their square. Some knew this was perimeter, but I explicitly avoided the word. Then we counted the number of boxes inside the square. Again, did not use the word area. 

Looking back, I will need to make sure that all students are measuring side length and perimeter with the same units here. I'll probably go with boxes just to avoid the mess and confusion that comes with the rulers. 

Tuesday, October 23, 2012

Day 35: Reflection

Unit 2 Assessments were handed back with feedback and an overall unit 2 grade (NOT a grade specifically for the assessment - something the kids are still struggling to accept). 

Students had the option to review their test and ask around for clarification, or continue to work on the puzzles from yesterday. A surprising number of students not only wanted more puzzles, but harder ones as well. Cool beans. 

Monday, October 22, 2012

Day 34: Logical Thinking

Both because I hadn't finished grading the Unit 2 Assessment yet, and because I hadn't really had time to get comfortable with how I wanted to start Unit 3 (Squares), I had students work on logic puzzles for the class period. My thinking was that instead of forcing formal proofs down students' throats, I'd rather just see them be able to thinking sequentially and justify their reasoning. Even with the angles in Unit 2, a LOT of students would label angles as "complimentary" without any reasoning for their choice. 

So I made 100+ copies of some free puzzles I found online and showed them the basics. The honors classes took to them like a fish to water, while the 5th hour class actually ended with a couple of referrals because of how many students simply refuse to do anything that requires independent thought. 

Overall, I'm happy I thought up a plan B that fit within my goals for the class. 

Friday, October 19, 2012

Day 33: Assessment

Unit 2 Assessment. Not much to report, other than my continual amazement at student behavior during a test. Both in complete inability to sit quietly, and in the expectation that test day is the day to finally ask for help. 

Because I'm a genius, I'm giving tests in 4 of my 5 classes today, so I'll have 130 exams to grade over a weekend in which I have absolutely no free time. 

Thursday, October 18, 2012

Day 32: Review & Practice

I made a point to get U2 Quiz 4 graded and back to students ASAP in preparation for the Unit 2 Assessment. SBG has really made me refocus how I spent my energy in terms of what I grade, and what I do when I'm up against a deadline like a formative assessment. In the past, if I was pressed for time, I would just grade quizzes after the fact. But why bother? If the quiz is meant to give targeted feedback to help guide studying, it's not worth anything if students take the test before seeing their results. 

We spent the day in the computer lab with students having the option to either practice with the online Carnegie software, look over their quiz results, or write their note sheet for the Unit 2 Assessment. 

Wednesday, October 17, 2012

Day 31: Assessment & Sketchpad

Hectic day. We took the 4th and final quiz (Angle Pairs) of the unit before the formative assessment. Afterwards, we rushed to the computer lab to work through an intro activity getting students familiar with Geometer's Sketchpad as we'll start using GSP as a discovery tool starting next week. 

So far, the quiz results are mildly disconcerting. Generally, students can classify angle pairs, but have no idea what to do with that information. They're easily flummoxed by algebraic expressions in place of discrete angle measurements. For example, saying "Angles 1 & 2 are a linear pair. Angle 1 = 50 deg and Angle 2 = ?" is solvable, but the exact same problem with Angle 1 = 2x+10 leaves students stumped. 

This confirms prior observations that even students who have been successful with math (specifically algebra) in the past have no real understanding of what they are doing and why they are doing it. I see the same phenomenon in physics with students who can graph anything in math class, but can't wrap their heads around the idea that 'y' is really just a label for the dependent variable which can be anything (same for 'x'). 

We spend so much time and effort drilling rote procedures into math students, that we (and more importantly, they) have lost all perspective as to our purpose. 

Tuesday, October 16, 2012

Day 30: Discussion

Whiteboard discussion of U2 WS4. With so many students (35), I've found that the best practice is giving each group 3-4 problems to whiteboard so that the entire worksheet is covered, with (hopefully) a little overlap. Rather than go over each board independently, I let students spend ~10 minutes walking around the room checking boards, taking notes and asking questions. Then we'll spend 10-15 mins as a group answering any lingering questions.

This works pretty well in my honors classes because most of the class will arrive having completed (or at least attempted) the homework. In my regular class, so many kids never attempt the practice, that we'd spend all hour just trying to whiteboard the problems and even then it would really only be 4-5 kids doing the work. Then the discussion is pointless, because the students who don't participate simply copy the work off the boards thinking that will be sufficient (which is obviously isn't). 

At it's best, I love this idea. It's dynamic - it gets the kids working together, moving around, and holding them responsible for seeking help. With an efficient class, we're easily doing 3-4 distinct things in a single class period which helps the day seem shorter. 

At it's worst, this strategy only benefits the kids who take it seriously. Of course, that's true with most methodologies  but sometimes I fear what students are telling parents who don't understand the modeling philosophy, especially when there aren't any pre-printed notes or a useful textbook to supplant the classroom environment. 

Monday, October 15, 2012

Day 29: Catchup

This was essentially a day for students to work on U2 WS4 in groups. It allowed my 5th & 6th Hours to catch up with my 2nd hour so that we'd all be on pace for Friday's Unit 2 Assessment. 

The main problem with allowing students time in class to work is how many of them will waste the time thinking that they "get it." This means they're not prepared to be productive in a class discussion the following day and they don't have any notes to use as a reference on open note quizzes. By the time feedback from the quiz arrives to indicate that they do not in fact "get it," they're short on time to fill in the gaps before a formative assessment. 

The related issue there is how many students simply refuse to think through problems on their own, and demand step-by-step instruction on every problem before they'll even take an assessment seriously. 

Friday, October 12, 2012

Day 28: Online Practice

Spent today in the computer lab working on the Carnegie Learning software. Students often get frustrated with how many problems the program makes them complete which I partially understand, but they refuse to acknowledge that pressing "hint" and guessing random answers tells the computer to assign more problems until mastery is achieved. Like most other work, students want me to walk them through every problem step by step until they reach a solution without any of their own effort. 

I do not require this work to be completed on any specific time table, but it is one of the components I mention to parents about why a student's grade might be what it is and I also might require completion before reassessments are administered. It only requires an internet connection, so students are free to work through the problems at home. 

We'll go over U2 WS4 on angle pairs on Monday and spend the rest of the week reviewing and preparing for the Unit 2 Assessment on Friday. 

Thursday, October 11, 2012

Day 27: Pairs of Angles (cont.)

Continuation of yesterday after a brief quiz on segment & angle addition. Assigned U2 WS4 on angle pairs which will be due Monday. 

Wednesday, October 10, 2012

Day 26: Pairs of Angles

2nd Hour is starting to move a little ahead of my other 2 classes, so I might focus on what they are doing to avoid getting confused. And because they don't make me rethink life choices I've made. 

So we went through different ways to pair up angles today. Sadly, it was more lecture oriented than I would typically strive for, but even when I lecture, it's not really a lecture. I just couldn't think of how to get students to "discover" the terms adjacent, complimentary, supplementary, etc. 

I used Geometer's Sketchpad (my favoritist thing in the whole world) to walk through examples and stop to document key terms. It makes it easy to show *why* the angle addition postulate (words that I never actually utter) arises from adjacent angles. I just speak in terms of "How could we write an equation to relate these three angles?" That way they're pressed to think instead of recall a certain postulate or theorem. 

We'll continue tomorrow with linear pairs and vertical angles, then practice using Carnegie on Friday. I expect to be done with Unit 2 early next week and have a formative assessment by next Friday. 

Tuesday, October 9, 2012

Day 25: Assessment and Discussion

First portion of the class was the 2nd quiz for Unit 2 on proper use of a ruler & protractor. It never ceases to amaze me how many students think during an assessment is the proper time to state "I don't know how to do this" and ask for direct instruction.

After the quiz we went over WS3 on classifying and naming angles. I made a specific point to emphasize the difference between those two words as well as the importance of evidence when declaring an angle to be "right." I really want students to understand the theme of the class as being the need justify reasoning without relying on an "I think..." mentality. 

The rest of the week will be spent on the last part of the unit which deals with identifying angle pairs and learning how to solve for their measure. This is another one of those brilliant ideas I had in terms of sequencing, but I never really got around to thinking about *how* to put it into action. Here's hoping for the best!

Monday, October 8, 2012

Day 24: Classifying Angles

More review than I had expected. I had originally come up with an idea to provide a bunch of angles of various sizes on laminated cards and have students sort them into groups. As many groups as they wanted and using whatever classification scheme they could come up with. We would then share approaches from different groups and lead the discussion to a consensus of ideas revolving around acute, obtuse, right, and straight. I was even prepared to allow the class to call those groupings whatever they wanted (big, small, perfect, line, whatever). 

But it turned out that angle classification is one of the few things that has been retained from prior math classes. So we quickly reviewed what the classifications are and moved directly into the practice with U2 WS3. 

Most students could work through the classification just fine, but working with angles and segments that have been broken into pieces was a challenge. Every year it is and every year it surprises me. Often, students will think that it's so easy they don't need to write an equation to solve for the unknowns, but when they're given more challenging problems they don't have *any* type of tested approach and will usually quit before they even try. 

I don't think students struggle setting up the equations once they see a few examples, but they do struggle with identifying their own limits and seeing that what might be easy today is about to get a heckuva lot more complicated and that our goal is to set in place skills & procedures that can be applied to ANY situation. 

Friday, October 5, 2012

Day 23: Measurement practice

Not much to report - the class was spent allowing students to work on U2 WS2 which dealt with using rulers and protractors correctly. 

I know a million different ways to show/explain this skill, but I have to find a consistent way to ensure that students are internalizing what we're doing. Generally, students think this is so basic that it's a waste of class time, but the truth is always revealed come quiz time. That might be an investigation to itself - how to work with students who consistently think they know material that they very obviously do not? 

I was a little optimistic that students seemed to connect our work with "angles as portions of a circle" to why protractors are half circles. A little disheartening to see so many use the ruler part of a protractor to measure an angle though. 

Thursday, October 4, 2012

Day 22: Measurement & Assessment

My students' philosophy in regards to quizzes and tests really needs an adjustment. I know their 9th grade math teacher gave "homework quizzes" probably 2-3x per week in lieu of grading actual homework, so it's really surprising that they are so unprepared for regular assessments. I allotted 15-20 mins for an 8 question quiz which looked *very* similar to the worksheet we've been dealing with all week on angles & proportions, and a lot of the students wanted extra time. 

Two issues at stake here: 

1) If you need more than 20 mins to answer 8 questions, you don't know the material. Extra time is not going to change all that. 

But this attitude is quite prevalent and something I've been thinking a lot about recently. My theory is that students are so trained to work recipes as procedures for doing problems in math & science that they honestly have no idea *what* they're doing. So they look at assessments as random - maybe their recipe will yield something fruitful, maybe it won't. If you offer a retake, they won't study, they'll just hope for "better" questions the second time around. 

In short, more time = maybe it will come to me.

2) "If I fail this quiz taken on the 4th day of new material, I'm going to fail the class." 

Both of these problems are related to mindset. Again, students are trained to think about assessments as worth points and if they fail, the just lost a bunch of points, never to be heard from again. That's why I love SBG so much - I could care less about points. Show me what you know. Do you think I expectation on Day 4 of a new unit? That's ridiculous. What's important is that we get it worked out before the end of the unit. 

I tried to explain to my classes my thoughts on the matter, but as per usual, they don't listen to much I say (hence modeling - have them experience what it's like to be ignored by people you're trying to help). I could see some heads nodding in agreement, but there is a vocal minority that hates the new system. They don't care about learning, progress, or growth. They just want a grade. 

But what is a grade if it doesn't reflect what you can (and can't) do?

Wednesday, October 3, 2012

Day 21: Discussion of Solving Proportions with Angles

I tried something new yesterday. U2 WS1 had about 20 problems broken into 4 groups - all related to proportions. The first group was very straightforward - solve the proportion. The 2nd group were word problems related to percents, so students had to set up the proportion themselves. The third group were word problems related to angles, so again, students were changed with the setup. And the last group had pictures of circles divided into pieces. Some amount of the circle was shaded in and I asked students to work through the steps given (fraction, proportion, angle) to connect everything.

It was way too much work to go over through a traditional discussion, so I had each class group whiteboard the same 4 problems (one problem from each section). When they were done, I gave them about 10 minutes to walk around the room checking everyone else's boards and asking questions. After that I tried to answer any lingering questions they might have had. 

It worked well in 2nd hour (I'm learning that most things work well in that class). 5th hour worked to the extent that students who were trying got something out of it, but that class is so frenzied, allowing them leeway to work in groups AND walk around class is like herding cats. 6th hour worked, but it took longer than expected, so we'll need a few minutes tomorrow to answer any questions. 

Proportions are again something that students learned in prior math classes, but the skills I'm seeing reinforce my theory that even students who can follow the steps involved have no real idea what they're doing. My point is to hammer home the idea that angles are portions of circles in the hopes that these students will not use angle and segment measurements interchangeably. 

Tuesday, October 2, 2012

Day 20: Practice with estimation

Piggy backing off yesterday's lesson and the idea that angles are a measure of rotation, I had my classes spend today getting comfortable with estimate angles through a couple of simple online games. 

Estimating Angles (from NRICH)

Alien Angles (from Math Playground)

The games are very simple, but I think that actually kept some students more engaged than they would have been otherwise. Some kids got tired of it pretty fast, but I reminded them of their homework (solving proportions) and the other online learning software we use (Carnegie). 

Much better day all around, but computer lab days usually are. I also handed back the results from Unit 1 with standard based progress report. Tomorrow we'll go over what it means and how to go about reassessments. 

Monday, October 1, 2012


I updated the overviews to Units 1 & 2. 

If there's request, I can upload the worksheets I've created so far.

Day 19: Rotational Measurements

I was really happy with my lesson plan today, but it was only something of a success in one of my three classes. Reasons why are more focused on student behavior which is not something I wanted to document here, so I'll focus on the ideas. 

In my experience teaching geometry, students have no real understanding for what an angle is. Asked to solve for an angle of a right triangle and they'll provide the length of a side (and vice versa). Asked to provide units for an angle and they'll say "feet." When I created this new curriculum over the summer, one of my primary goals was the importance of angles as measures of rotation. 

So I started class with a demo involving 8 students in a circle around me. I used meter sticks to divide up the circle and talked about fractions. From there I expand the idea of fractions to describe how far I would turn to face certain people. And finally I had the circle enlarged to emphasize the idea that rotational measures are independent of distance measures. 

Then we discussed distance measures and their units, since most students are familiar with those ideas. We then went back to rotational measures and probed the idea of units. Some students suggested percents (which was awesome), but it took a LOT of probing to get the students to say "degrees" (exactly my point). They've all heard of angles & degrees before, but had seemingly never connected those terms to the idea of rotation. 

We then worked through some examples of setting up proportions using first fractional turns, then degrees in a circle, and finally converting to percents.

I had a nice side story prepared about why there are 360 degrees in a circle, but we were unable to get to it in two of the classes today. When simply asked about it, most students said "because 1/4 turn is a right angle which is 90 deg, so 4 * 90 = 360." Ok, so where did 90 deg come from? At that point, they were happy to reason in circles without seeing the missing explanation of where 360 even comes from. Oh well.