# #MakeoverMonday - Sun Room Carpet

This is my attempt to revise the textbook problem Dan Meyer posted here:

## What I did:

• Bring the most interesting part of the problem (to me) to the front of the task. The idea of requiring the seams to go in the same direction with a minimum number of seams is the only opening for multiple answers in this problem. Start with this, and the students will already have a chance to disagree about answers, which we know is a good way to get conversations going.
• Frame this possibility of multiple arrangements visually, not with text. By placing the carpet strips in different configurations when first displaying the task, I'm nudging students toward different answers. Again, this conflict is important to spicing up a pretty plain textbook task.
• Get rid of those mixed metric/English units immediately. There is no reason that a person would measure the room in meters and then deal with a store that sells in feet and yards. There's enough here to keep things interesting without dealing with meters and yards.
• Leave out unnecessary vocabulary like 'bolt' and 'seam'. One more move to reduce the text overload of the problem.

Here's the rundown of the lesson:

Hand out slips of paper with either Situation A or Situation B shown below.

Once students have drawn their lines, share a few of the drawings to show the differences. Pose the question: Which situation (A or B) will require more cutting?

Have students make and record their guesses.

Move to the second act here. What information would you need to answer the question precisely?

There's some wiggle room here for what happens next. If the students ask for the measurements, you could give them this diagram:

In reality though, students could figure this out just by making measurements with a ruler on the diagram. No big deal either way.

We then ratchet up the task by bringing up the tape factor:

The students might not realize that the carpet is attached at the edges of the room (though not usually with tape) which is why it's important to bring up this point. At this point the dimensions diagram still might be requested, but it isn't really necessary until students start reporting their answers. What units are involved?

It would be at this stage when I would actually ask the students which situation is the better option for carpeting the room. The ambiguity is on purpose - students need to decide what it really means for a situation to be better. Is it cost? Time required to cut? Amount of tape? All of these factors come into play. The original problem ultimately asks what the total cost is for the carpet and the tape, but there are lots of possibilities for what can be done at this point in the lesson. Here are some possibilities:

At the hardware store, you learn the following:

• A 30 foot roll of double sided tape costs \$4.85
• The carpet sells for \$22.95 per square yard
• What is the minimum cost to carpet the room?

Or, building on that information:

Suppose you also make \$11.50 per hour to lay carpet.

Which situation do you choose to maximize the money you make on the job? What information would you need to answer this question?

Having students work to answer this efficiently means getting students that have worked on the two situations to talk to each other.

Follow up questions to throw in:

• If you had a choice of 3, 5, or 12 foot wide strips of carpet, which would result in the cheapest overall job
• If you can cut the carpet at 2.5 inches/second for a straight cut, with an additional second for each cut, how long would it take for you to prep the carpet for the room?

This is my first attempt in the Make over Monday series, and I'm exhausted. Also, this was fun. What's next, Dan?

# Tea with Lee Magpili and the LEGO Mindstorms EV3

Though my schedule being back in the US this summer has been busy, when I learned that Lee Magpili was going to be in town, I cleared my schedule. I first met Lee when I was working with the Bronx FIRST LEGO League initiative several years ago. He was a quiet presence in comparison to the energetic middle school students that attended our workshops to play with LEGO robots, but I quickly learned of his prowess with building with LEGO elements. His rovers navigated the FLL field with ease and used mechanisms that balanced simplicity with effectiveness. Eventually he mentored an FLL team to do exceptionally well. Like all great FLL coaches, however, he insisted on the students doing the work. In our conversations at that time, I quickly understood that Lee believed (and continues to believe) that LEGO is an amazing platform upon which to learn an enormous range of useful skills. Robotics, in particular, capitalizes on the unique blend of play and learning through LEGO to get students to understand the engineering design process. Lee is a believer in the potential for students to be quickly engaged and motivated to work hard when the right tools are around.

It was consequently no surprise when I learned Lee had been selected for a job with LEGO education in Denmark a couple of years ago. He and I wrote back and forth periodically about the position and what it entailed, but for a while our conversations turned noticeably away from the details of his work. I figured this was just a consequence of the distance and I left it at that.

This ended last January with the announcement of the LEGO Mindstorms EV3. When Lee posted a link to the announcement on Facebook, I suddenly understood. It made me realize that like any good designer, he kept his ideas secret until they were ready to share with the world. (I assume a pretty airtight NDA was also involved.)

Lee sat down with me at Saints Alp Teahouse in New York for some bubble tea, snacks, and conversation about the EV3. What struck me was that Lee's enthusiasm for using LEGO as a learning tool hasn't just been maintained, it has grown considerably since becoming part of the EV3 team. As you might also expect, he was excited to show me the bits and pieces of the kit that will be coming out in August.

From a LEGO designer's perspective, the attention to detail in acknowledging the desires of the LEGO fan community and the limitations of the NXT set will most definitely be appreciated. There are some subtle changes that made me excited given my own experiences building with the curves of the NXT and its parts.

For example, a reshaping of the motor has made it much easier to attach pins and secure it to designs:

Sensors can be attached using a single pin if needed:

I also suspect that many people will discover ways that alignment between different components will be much easier with the new set:

Lee also spoke a lot about the care that he and the team have taken to make the bar for entry with the kit low, and the ceiling high. The education kit will include instructions for building modules that can be used in different designs. A conveyor belt doubles as a set of tracks. A motor-wheel module can be built that is sturdy but easy to build upon. This will help students (and teachers) minimize the frustration that inevitably occurs when straying from build instructions to pursue an idea for a new design. The strengths associated with building with Technics parts will be a lot more intuitive to newcomers that may have only worked with bricks.

I am excited to get my hands on one of these kits. In my robotics class this year, students grew considerably in their ability to conjure up a design and make it happen with the bricks. Students often got frustrated by the curves of the NXT motors getting in the way of their designs. The ease of attaching motors directly to the programmable brick of the EV3 will make it even easier to get students learning programming techniques. The on brick features for prototyping and programming will make things much easier for trying out quick ideas, especially on an FLL field.

It was good catching up with Lee - he is a person to watch in the world of LEGO Education. He was at the FIRST World Festival to demonstrate the EV3 to FIRST LEGO League teams, not to mention members of the Board of Directors at the LEGO group. He told me that his plans include photographing Gyro Boy in Times Square and Washington Square Park. Though he assures me that the robot named 'Evan' that has been touring the world to demonstrate the EV3 is not named after me, I'm going to continue to assume that it is.

# 2012-2013 Year In Review – Learning Standards

This is the second post reflecting on this past year and I what I did with my students.

My first post is located here. I wrote about this year being the first time I went with standards based grading. One of the most important aspects of this process was creating the learning standards that focused the work of each unit.

### What did I do?

I set out to create learning standards for each unit of my courses: Geometry, Advanced Algebra (not my title - this was an Algebra 2 sans trig), Calculus, and Physics. While I wanted to be able to do this for the entire semester at the beginning of the semester, I ended up doing it unit by unit due to time constraints. The content of my courses didn't change relative to what I had done in previous years though, so it was more of a matter of deciding what themes existed in the content that could be distilled into standards. This involved some combination of concepts into one to prevent the situation of having too many. In some ways, this was a neat exercise to see that two separate concepts really weren't that different. For example, seeing absolute value equations and inequalities as the same standard led to both a presentation and an assessment process that emphasized the common application of the absolute value definition to both situations.

### What worked:

• The most powerful payoff in creating the standards came at the end of the semester. Students were used to referring to the standards and knew that they were the first place to look for what they needed to study. Students would often ask for a review sheet for the entire semester. Having the standards document available made it easy to ask the students to find problems relating to each standard. This enabled them to then make their own review sheet and ask directed questions related to the standards they did not understand.
• The standards focus on what students should be able to do. I tried to keep this focus so that students could simultaneously recognize the connection between the content (definitions, theorems, problem types) and what I would ask them to do with that content. My courses don't involve much recall of facts and instead focus on applying concepts in a number of different situations. The standards helped me show that I valued this application.
• Writing problems and assessing students was always in the context of the standards. I could give big picture, open-ended problems that required a bit more synthesis on the part of students than before. I could require that students write, read, and look up information needed for a problem and be creative in their presentation as they felt was appropriate. My focus was on seeing how well their work presented and demonstrated proficiency on these standards. They got experience and got feedback on their work (misspelling words in student videos was one) but my focus was on their understanding.
• The number standards per unit was limited to 4-6 each...eventually. I quickly realized that 7 was on the edge of being too many, but had trouble cutting them down in some cases. In particular, I had trouble doing this with the differentiation unit in Calculus. To make it so that the unit wasn't any more important than the others, each standard for that unit was weighted 80%, a fact that turned out not to be very important to students.

### What needs work:

• The vocabulary of the standards needs to be more precise and clearly communicated. I tried (and didn't always succeed) to make it possible for a student to read a standard and understand what they had to be able to do. I realize now, looking back over them all, that I use certain words over and over again but have never specifically said what it means. What does it mean to 'apply' a concept? What about 'relate' a definition? These explanations don't need to be in the standards themselves, but it is important that they be somewhere and be explained in some way so students can better understand them.
• Example problems and references for each standard would be helpful in communicating their content. I wrote about this in my last post. Students generally understood the standards, but wanted specific problems that they were sure related to a particular standard.
• Some of the specific content needs to be adjusted. This was my first year being much more deliberate in following the Modeling Physics curriculum. I haven't, unfortunately, been able to attend a training workshop that would probably help me understand how to implement the curriculum more effectively. The unbalanced force unit was crammed in at the end of the first semester and worked through in a fairly superficial way. Not good, Weinberg.
• Standards for non-content related skills need to be worked in to the scheme. I wanted to have some standards for year or semester long skills standards. For example, unit 5 in Geometry included a standard (not listed in my document below) on creating a presenting a multimedia proof. This was to provide students opportunities to learn to create a video in which they clearly communicate the steps and content of a geometric proof. They could create their video, submit it to me, and get feedback to make it better over time. I also would love to include some programming or computational thinking standards as well that students can work on long term. These standards need to be communicated and cultivated over a long period of time. They will otherwise be just like the others in terms of the rush at the end of the semester. I'll think about these this summer.

You can see my standards in this Google document:
2012-2013 - Learning Standards

I'd love to hear your comments on these standards or on the post - comment away please!

# Editing Khan

Let's be clear - I don't have a problem with most of the content on Khan Academy. Yes, there are mistakes. Yes, there are pedagogical choices that many educators don't like. I don't like how it has been sold as the solution to the educational ills of our world, but that isn't my biggest objection to it.

I sat and watched his series on currency trading not too long ago. Given that his analogies and explanations are correct (which some colleagues have confirmed they are) he does a pretty good job of explaining the concepts in a way that I could understand. I guess that's the thing that he is known for. I don't have a problem with this - it's always good to have good explainers out there.

The biggest issue I have with his videos is that they need an editor.

He repeats himself a lot. He will start explaining something, realize that he needs to back up, and then finishes a sentence that hadn't really started. He will say something important and then slowly repeat it as he writes each word on the screen.

This is more than just an annoyance. Here's why:

• One of the major advantages to using video is that it can be good instruction distilled into great instruction. You can plan ahead with the examples you want to use. You can figure out how to say exactly what you need to say and nothing more, and either practice until you get it right, or just edit out the bad takes.
• I have written and read definitions word by word on the board during direct instruction in my classes. I have watched my students faces as I do it. It's clearly excruciating. Seeing that has forced me to resist the urge to speak as I write during class, and instead write the entire thing out before reading it. Even that doesn't feel right as part of a solid presentation because I hate being read to, and so do my students. This doesn't need to happen in videos.
• If the goal of moving direct instruction to videos is to be as efficient as possible and minimize the time students spend sitting and watching rather than interacting with the content, the videos should be as short and efficient as possible. I'm not saying they should be void of personality or emotion. Khan's conversational style is one of the high points of his material. I'm just saying that the 'less is more' principle applies here.

I spent an hour this morning editing one of the videos I watched on currency exchange to show what I mean. The initial length of the video was 12:03, and taking out the parts I mentioned earlier reduced it to 8:15. I think the result respects Khan's presentation, but makes it a bit tighter and focused on what he is saying. Check it out:

The main reason I haven't made more videos for my own classes (much to the dismay of my students, who really like them) is my insistence that the videos be efficient and short. I don't want ten minute videos for my students to watch. I want two minutes of watching, and then two or three minutes of answering questions, discussing with other students, or applying the skills that they learned. My ratio is still about five minutes of editing time for every minute of the final video I make - this is roughly what it took this morning on the Khan Academy video too. This is too long of a process, but it's a detail on using video that I care too much about to overlook.

What do you think?

# 2012-2013 Year In Review - Standards Based Grading

This is the first in a series of posts about things I did with my classes this year.

When I made the decision last fall to commit to standards based grading, these were the main unknowns that hung at the back of my mind:

• How would students respond to the change?
• How would my own use of SBG change over the course of the year?
• How would using SBG change the way I plan, teach, and assess?

These questions will all be answered as I reflect in this post.

### What did I do?

In the beginning of the year, I used a purely binary system of SBG - were students proficient or not? If they were proficient, they had a 5/5. Not yet proficient students received a 0/5 for a given standard. All of these scores included a 5 point base grade to be able to implement this in PowerSchool.

As the semester went on, the possible proficiency levels changed to a 0, 2.5, or 5. This was in response to students making progress in developing their skills (and getting feedback on their progress through Blue Harvest but not seeing visible changes to their course grade. As much as I encouraged students not to worry about the grade, I also wanted to be able to show progress through the breakdown of each unit's skills through PowerSchool. It served as a communication channel to both parents and the students on what they were learning, and I could see students feeling a bit unsatisfied by getting a few questions correct, but not getting marked as proficient yet. I also figured out that I needed to do more work defining what it meant to be proficient before I could really run a binary system.

By the start of the second semester, I used this scheme for the meaning of each proficiency score:

• 1 - You've demonstrated basic awareness of the vocabulary and definitions of the standard. You aren't able to solve problems from start to finish, even with help, but you can answer yes/no or true or false questions correctly about the ideas for this standard.
• 2 - You can solve a problem from start to finish with your notes, another student, or your teacher reminding you what you need to do. You are not only able to identify the vocabulary or definitions for a given skill, but can substitute values and write equations that can be solved to find values for definitions. If you are unable to solve an equation related to this standard due to weak algebra skills, you won't be moving on to the next level on this standard.
• 3 - You can independently solve a question related to the standard without help from notes, other students, or the teacher. This score is what you receive when you do well on a quiz assessing a single standard. This score will also be the maximum you will receive on this standard if you consistently make arithmetic or algebraic errors on problems related to this standard.
• 4 - You have shown you can apply concepts related to this standard on an in-class exam or in another situation where you must identify which concepts are involved in solving a problem. This compares to success on a quiz on which you know the standard being assessed. You can apply the content of a standard in a new context that you have not seen before. You can clearly explain your reasoning, but have some difficulty using precise mathematical language.
• 5 - You have met or exceeded the maximum expectations for proficiency on this standard. You have completed a project of your own design, written a program, or made some other creative demonstration of your ability to apply this standard together with other standards of the unit. You are able to clearly explain your reasoning in the context of precise mathematical definitions and language.

All of the standards in a unit were equally weighted. All units had between 5 and 7 standards. In most classes, the standards grade was 90% of the overall course grade, the exception being AP Calculus and AP Physics, where it was 30%. In contrast to first semester, students needed to sign up online for any standards they wanted to retake the following day. The maximum number of standards they could retake in a day was limited to two. I actually held students to this (again, in contrast to first semester), and I am really glad that I did.

Before I start my post, I need to thank Daniel Schneider for his brilliant post on how SBG changes everything here. I agree with the majority of his points, and will try not to repeat them below.

### What worked:

• Students were uniformly positive about being able to focus on specific skills or concepts separate from each other. The clarity of knowing that they needed to know led some students to be more independent in their learning. Some students made the conscious decision to not pursue certain standards that they felt were too difficult for them. The most positive aspect of their response was that students felt the system was, above all else, a fair representation of their understanding of the class.
• Defining the standards at the beginning of the unit was incredibly useful for setting the course and the context for the lessons that followed. While I have previously spent time sketching a unit plan out of what I wanted students to be able to do at the end, SBG required me not only to define specifically what my students needed to do, but also to communicate that definition clearly to students. That last part is the game changer. It got both me and the students defining and exploring what it means to be proficient in the context of a specific skill. Rather than saying "you got these questions wrong", I was able to say "you were able to answer this when I was there helping you, but not when I left you alone to do it without help. That's a 2."
• SBG helped all students in the class be more involved and independent in making decisions about their own learning. The strongest students quickly figured out the basics of each standard and worked to apply them to as many different contexts as possible. They worked on communicating their ideas and digging in to solve difficult problems that probed the edges of their understanding. The weaker students could prioritize those standards that seemed easiest to them, and often framed their questions around the basic vocabulary, understanding definitions, and setting up a plan to a problem solution without necessarily knowing how to actually carry out that plan. I also changed my questions to students based on what I knew about their proficiency, and students came to understand that I was asking a level 1 question compared with a level 3 question. I also had some students giving a standards quiz back to me after deciding that they knew they weren't ready to show me what they knew. They asked for retakes later on when they were prepared. That was pretty cool.
• Every test question was another opportunity to demonstrate proficiency, not lose points. It was remarkably freeing to delete all of the point values from questions that I used from previous exams. Students also responded in a positive way. I found in some cases that because students weren't sure which standard was being assessed, they were more willing to try on problems that they might have otherwise left blank. There's still more work to be done on this, but I looked forward to grading exams to see what students did on the various problems. *Ok, maybe look forward is the wrong term. But it still was really cool to see student anxiety and fear about exams decrease to some extent.

### What needs work:

• Students want more detail in defining what each standard means. The students came up with the perfect way to address this - sample problems or questions that relate to each standard. While the students were pretty good at sorting problems at the end of the unit based on the relevant standards, they were not typically able to do this at the beginning. The earlier they understand what is involved in each standard, the more quickly they can focus their work to achieve proficiency. That's an easy order to fill.
• I need to do more outreach to parents on what the standards mean. I thought about making a video at the beginning of the year that showed the basics, but I realize now that it took me the entire year to understand exactly what I meant by the different standards grades. Now that I really understand the system better, I'll be able to do an introduction when the new year begins.
• The system didn't help those students that refuse to do what they know they need to do to improve their learning. This system did help in helping these students know with even more clarity what they need to work on. I was not fully effective in helping all students act on this need in a way that worked for them.
• Reassessment isn't the ongoing process that it needs to be. I had 80 of the 162 reassessment requests for this semester happen in the last week of the semester. Luckily I made my reassessment system in Python work in time to make this less of a headache than it was at the end of the first semester. I made it a habit to regularly give standards quizzes between 1 or 2 classes after being exposed to the standard for the first time. These quizzes did not assess previous standards, however, so a student's retake opportunities were squarely on his or her own shoulders. I'm not convinced this increased responsibility is a problem, but making it an ongoing part of my class needs to be a priority for planning the new year.

I am really glad to have made the step to SBG this year. It is the biggest structural change I've made to my grading policy ever. It led to some of the most candid and productive conversations with students about the learning learning process that I've ever had. I'm going to stop with the superlatives, even though they are warranted.

# Milestones at the start of summer: A tribute

I used this LEGO car in a five minute demo lesson - my first lesson ever - on Newton's laws of motion. It was a gimmick to get the people in the room thinking about what they knew about forces, and served this purpose perfectly. This was in the beginning stages of my decision during my senior year at Tufts to pursue teaching rather than engineering after graduation.

It sat on the bookshelf next to my desk in both of my New York City apartments. It made its way into a suitcase that a friend took to Zambia. It was one of the items that I took out of the storage last summer with a smile, and was among the knick-knacks that didn't get tossed in the move to the apartment in Hangzhou for next year.

This LEGO car rolled across the floor of the new apartment last week, the final week of my tenth year teaching. It made me think back to the many adventures that have been my life ever since I received my acceptance letter to the New York City Teaching Fellows program in 2003. I worked with an incredible group of teachers in the Bronx for seven years, helped open the KIPP NYC College Prep high school, and then made the move to Hangzhou where I have enjoyed teaching kids and working with some fantastic folks from all over the world.

Even though it is the start of summer vacation, my head is still very much in the teaching game. It's gratifying to know that I can reinvent myself every year after a summer of reflection and meditation on what went well and what did not. I am motivated by my students comments in end-of-year surveys that my enthusiasm for learning and sharing new things gets them excited to be in the classroom with me. The unique experience of working with teenagers compels me to still devote energy and time to making myself better at what I do.

To the students that I have worked with over the past ten years: thank you for giving me the most exhilarating, satisfyingly unpredictable, and meaningful ten years I never knew I wanted in a career. To my colleagues: thank you for teaching me what it means to work hard for the right reasons and toward the right ends. To my family: thank you for supporting me in all that I do.

Have a great summer everyone!

# Three Acts - Counting with dots and first graders

I had an amazing time this afternoon visiting my wife's first grade class. I've been talking forever about how great it is to take a step out of the usual routines in class and look at a new problem, and my wife invited me in to try it with her students.

Here's the run-down.

### Act 1

Student questions (and the number of students that also found the questions interesting):

• Why do the dots come together? (8)
• Why are the dots making pictures and not telling us what they mean? (8)
• Why are some dots going together into big dots, and others staying small? (13)
• Why do some of the dots form blue lines before coming together?

My questions (and the number of students that humored me):

• How many dots are there at the end? (8)
• What is the final pattern of dots after the video ends? (11)

Guesses for the number of dots ranged from a low of 20 to a high of 90.

### Act 2

What information did they want to know?

• They wanted to see the video again.
• Seven students asked about the numbers of tens or ones in each group. (I jumped on the use of that vocabulary right away - it seemed they are comfortable using this vocabulary based on my conversations with them.)
• I showed them the video and gave them this handout since I didn't have video players for all of the students:
grouping dots

What happened then was a series of amazing conversations with some really energetic and enthusiastic kids. They got right to work organizing and figuring out the patterns.

### Act 3

We watched the video and discussed the results and how they got their answers. Lots of great examples of student-created systems for keeping track of their counting. We then watched the Act 3 video:

While nobody had the total number correct, I was quite impressed with their pride in being close. More interesting was how little they cared that they didn't get the exact answer. I asked who was between 70 and 80, and a few kids raised their hands, and then the same with 50 - 70. One student was one off. Most were within ten or so of the correct answer. The relationship between the guesses and their answers after analysis was something we touched upon, but didn't discuss outside of some one-on-one conversations.

The absolute highlight of the lesson was when I asked why they thought nobody had the exact answer. One student walked up to the projector screen with out hesitation and pointed here:

She said "this is what made it tough" and then sat back down.

We had a little more time, so we watched a sequel video:

I asked what they saw that was different aside from the colors. One student said right away that he figured it out, the same student that first shouted out 'tens!' in Act 1. We lacked the time to go and figure it out, so we left it there as a challenge to figure out for the next class.

Footnotes:

• Any high school or middle school math teacher that wants to see how excited students can be when they are learning math needs to go take a group of elementary students through a three act. I wish I had done this during the dark February months when things drag for me. My wife asked me to do this to see how it works, but I think I got a lot more enjoyment out of the whole experience.
• I made a conscious decision not to include any symbolic numbers in this exercise. It adds an extra layer of abstraction that takes away from the students figuring out what is going on. I almost put it back in when I wasn't sure whether it was obvious enough. I am really glad I left it out so the students could prove that they didn't need that crutch.
• This is written in Javascript using Raphael. You can see a fully editable version of the code in this JSFiddle.
• All files are posted at 101 Questions in case you want to get the whole package.