Which problems should we go over? An impromptu graphing activity in AP Statistics

dot-plot-cropped.jpgThis week we were reviewing for the first unit test in AP Statistics, and I stumbled upon an in-class activity that highlights the difference between bar charts and histograms, particularly in the confusing situation where categorical data is masquerading as quantitative data.  Plus it grew completely organically from our need to organize our review questions.

To prepare for our unit review, the homework I assigned was for students to browse through the 40 homework problems at the end of the unit and pick three that they particularly wanted to go through in class.  This seemed like a good idea, until the next morning when I was getting ready for class, and I realized how hard it was going to be to have a whole class reach consensus on which  problems to go over. In fact, it was going to be a challenge to even record which problems were most popular. I thought about doing a google poll, but that required more setup than I had time for.  So I just went to the board and drew this  axis (painstakingly recreated because I didn’t think to take a picture at the time):


What is wrong with this picture? Post your answer in the comments….

Then I told the students to go to the board and mark the problems they had chosen with a dot, to make a dot-plot while I took attendance, figuring I would know which problems students wanted to review.  First three rules of dealing with data: make a picture, make a picture, make a picture.  This is what they drew:Dot plot cropped

I was trying to decide what the cut-score should be for a problem. Clearly I didn’t have time to go over all of the problems that students wanted to review, and equally clearly, numbers 26 and 17 were must-dos. But I wondered how many requests would justify class time spent, and started thinking about the statistics of the situation.

While thinking about this, I asked the class, “What does the distribution look like?”  Short silence, followed by some tentative “Multi-modal” and “Skewed” answers.  So I asked another question: “What kind of data is this?” (Yes, I know. Grammar.  But that’s what I asked.)

The class confidently described it as quantitative.  One student then looked dubious, and said, “Wait, no, it’s categorical.”  Some argument ensued.  So I suggested that instead of going over 26 and 18, maybe we should just average them and go over 22.  They immediately saw that the problem number carried no mathematical information, and that the number was just an identifier in this case.

I asked, “But can we make a histogram? Sometimes a dot plot is just a discrete histogram, but not in this case. Can we use these same data and make a proper histogram?”

It took awhile, but a few of the students suggested that the horizontal axis could be “number of students requesting the problem,” and then we were off.  We made a table of values:

Table of values

And plotted the histogram on our calculators:


Then we could describe the shape of the distribution (Unimodal and strongly skewed right).  We calculated the Min (zero), Q1 (zero), Median (zero!) and Q3 (two), got an IQR of 2, and found that there were two outliers at 9 and 6 requests each, which corresponded to questions 26 and 18.  I said, “Great, then we only have to go over two questions!”

After a few seconds of silence, one student timidly put up her hand and said, “I know it’s not an outlier, but can we go over number 31 also?” (Sometimes they don’t realize when I’m joking….)

Side notes:

  • While going over question 26, I erased half of our dot plot, which gave us a great illustration of the information you lose when making a histogram; once I erased the dotplot, I didn’t have a record of the problem numbers anymore.  Doh!
  • When we looked at our summary statistics (1-var stats on our table of values in the TI-84), we discovered a mistake in our initial graph. Post in the comments if you see what I did wrong!

Bonus side note: My principle dropped in to observe my class while we were doing this.  He stopped by at the end of the day to tell me that watching this activity had been a highlight of his day.  Stats FTW!



Can you teach this?

This semester, for the first time ever, all of my classes are honors or AP classes.  For the most part, the students are bright, engaged, and concerned about doing well.  But I have discovered that many of these high-end learners have weak study skills.

When I asked how many of them take notes, the students all looked at each other, confused, and most of them tentatively raised their hands.  I clarified.

“I mean, either during class, or when you are reading your textbook at home—either one.”

The hands stayed up, but no one looked very comfortable.  So I tried a different question.

“How many of you feel really awkward taking notes, because you have no idea what you are supposed to be writing down and why?”

I had touched the heart of the matter. Every hand in the class shot up, and everyone started talking.

About two thirds of the students claim to take notes, but apparently not one of them does so with a clear, internal goal in mind.  These are kids who are about to go to college, where I fervently hope they will step outside their intellectual comfort zones, struggle, and need to take and use good notes.  But most of these same kids have never had to do that before. They are bright, quick learners, and all through their school careers, they have grasped things faster than their classmates.  They have never needed to take notes, or study, or break tasks down into smaller pieces, or seek extra help on anything. They know they are supposed to develop study skills, and many of them are poking about, trying different techniques in a meandering, haphazard, and completely tentative way.

In short, their executive functioning is underdeveloped.  At least I think that’s the problem; I’m not a psychologist, and since I came to teaching on a non-traditional path, I didn’t even take a child psychology class in college.  Everything I know about executive function I have acquired obliquely, through discussions in faculty meetings and special-ed plan reviews, and in blogs I have read while worrying about my own two kids.  But I went over to the Harvard Center on the Developing Child to see what they had to say.  Here’s the first paragraph on their  “Executive Functioning and Self-Regulation” page:

Executive function and self-regulation skills are the mental processes that enable us to plan, focus attention, remember instructions, and juggle multiple tasks successfully. Just as an air traffic control system at a busy airport safely manages the arrivals and departures of many aircraft on multiple runways, the brain needs this skill set to filter distractions, prioritize tasks, set and achieve goals, and control impulses.

This seems to me to be the issue. At our school, we spend a great deal of time talking about executive function in our students with learning disabilities or behavioral issues, but I practically NEVER address it explicitly for the highest-end learners.  Oh, I tell them they should do their homework, or take notes, or study for a test, but those directions are usually not very effective, and I don’t really address the problem of building up executive function as a whole.  So that’s my goal this year in AP Statistics. I have several strategies, but the one I’m most excited about is making the students help teach the class.

Every day I assign reading in our textbook (Stats, Modeling the World, by Bock, Velleman and De Veaux).  In past years, I think the students have mostly done the reading, if by “doing the reading” I mean “sitting with the book open, and passing their eyes over the words on the white parts of the page.”  Many of them skip all the special boxes, callouts and visual cues that the textbook writers helpfully inserted. It’s sad, because in this book those marginal bits are important, well-structured, and often very funny.  But the students have been trained by years of really poor textbooks with irrelevant pictures and cutesy mnemonics, and they “know” that most of the stuff in the book can be safely ignored.  They grant the marginal boxes about as much attention as they give to ads on a website.  They skim the main text, and then they come to class, ready for me to “teach them” what they have just read.

So how do I get them to read the book skillfully, teach themselves as much as they can, and come to class ready to pump each other (and me) for the information they still need? Here’s my plan:

At the beginning of class (an 80-minute block), I will randomly pick two students by cluster sampling, to present a ten-minute lesson on the previous night’s reading.  Once I announce the pair, they will have ten minutes to review their notes and put together an organized presentation with the following three main parts: Key new concepts, connection to previous material, and points of confusion.  Everyone in the class who is NOT presenting will have those ten minutes to review their own notes,  formulate questions, predict points of confusion, and prepare to be helpful.  I’m trying to maintain the zen practice of “don’t know mind” about how this will play out in class, but I’m struggling; I am very attached to the beautiful script in my head. With a meaningful reason to read and take notes, students will be prepared and engaged, and the entire class will attain instant enlightenment.  Right?

Of course the universe is under no obligation to follow my script. I’d like to start with a good, structured foundation, and of course be ready to adjust as it plays out in the classroom. Can any of you out there help me anticipate potential problems and avoid them? Does anyone see a wonderful opportunity for student learning that I may be missing because of the structure of this plan?  If so, please leave advice in the comments section below.



Visibly Random Grouping, one semester in.


This year I tried something new: visibly random grouping (VRG) every single day. I posted my plan back in September, full of optimism and backed by research.  In short, I planned to assign students  randomly to groups every day, using a deck of cards that I shuffle ostentatiously in front of them.  My classroom has 6 large tables, and my largest class has 24 students, so I assigned aface value to each table and sorted out custom decks for each class.IMG_0386

In anticipation of the new year, I bought decks of cards and a spiffy little card holder.  I prepared myself for pushback from the students. I even prepared for pushback from parents, fearing that students with learning or anxiety issues might find the social uncertainty too  stressful and might complain at home.

Now, with four days left in the term before finals, I am trying to process my thoughts about the technique, so I can decide whether to continue it next semester, and how I might plan to modify it.

The things I feared, and how they played out:

  • Students refusing to sit at assigned tables. This ended up not being much of an issue. Maybe my students were unusually compliant, but they didn’t usually argue to sit with their friends. As the term went on, I had a few students who would ask to sit alone in one of the satellite “testing desks” (designated by jokers in the separate “test-day deck” I put together) when we weren’t doing group work, and I sometimes allowed it. I think they were self-identifying as being easily distracted by the students around them.
  • Students switching cards on the sly. This happened some during the first week. It stopped when I started handing them their cards face up, so they could see I knew which card they got.  Sometimes it still happens occasionally, but it’s not a huge problem, and when I notice a statistically unlikely trend, I pay closer attention.
  • Random match-ups failing spectacularly.This happened, but not in the class I expected. In my lower level class, where I expected the most problems, the randomness only occasionally generated bad combinations.  However, in my Honors Geometry class (the largest class, and mostly freshmen), I ended up having to assign seats at separate tables to six of the kids for the long term, and randomly grouping the rest of the class around them. Those six just couldn’t keep from talking when sitting together in any combination.
  • Difficulty remembering names. Yep, it was hard. But I persevered and learned them after a week or so.

The things I hoped for, and how they played out:

  • Better classroom culture. Well, the statistician in me doesn’t dare draw conclusions from a non-blinded, uncontrolled study.  But I did get anecdotal feedback from a classroom observer who asked what was up with the cards, and a student told her how much she liked it.  The student said she probably never would have learned all the other students’ names if she hadn’t been forced to sit with them all over time. N of one—check.  I’ve not had many behavior issues this semester, but maybe the kids are just more compliant, or I’m getting better at classroom management.
  • Fewer entrenched bad behaviors. Again, like above, I can’t tell for sure, but I didn’t have as much trouble early on with students talking off-task and needing redirection.  As the students all got to know each other better, the behaviors did emerge, but again, I don’t think it was as bad as it has been for me in previous years.
  • More transfer of knowledge. One of the arguments for this method in the literature is that it creates more transfer of knowledge from student to student, and less reliance on the teacher. I am not sure that I’ve seen that. But then, I am not that good yet at designing group work, and just randomizing seating isn’t going to change that.  We’ll see how this develops as I get more proficient at collaborative learning.

Things that happened that I never thought about:

  • Playing with the cards instead of getting ready for class. If you give it to them, they will play with it.  Students devised ingenious games, playable with only three or four cards. They would gather cards from other tables for more intricate games.  They would build houses of cards. I needed to get the cards back quickly, or I had trouble gaining their attention to start class. The Honors students were worst about this.
  • Damaging the cards. Corollary to the above: If you give it to them, they will destroy it.  Idly tearing the cards, spilling things on them, coloring in the spots, folding the corners–you name it, I have watched students do it.  Really, guys?  You can’t have a card for three minutes without defacing it? Again, the Honors students were the worst about this.


  • Difficulty handing back papers. I didn’t realize how helpful it was that students usually sat in the same places, when it came to handing back papers. When I didn’t randomize, I got very quick at locating students, but when they sit randomly I have to search for each and every one. It really does take a lot more time.
  • Knot of students around my desk before every class. Students mob my desk when they come in early, and they want me to stop what I am doing to pass out the cards. I tried letting them take their own, or assigning a shuffler, but it led to more sneaky seat-swapping. It makes it difficult to do last minute class prep, especially between blocks.

Help me!

I think I’m going to do it again next semester so I can see if the benefits continue with different students. I do wish I had some kind of instant, electronic card-dealing program that would allow me to select a partial deck and deal cards to a list of students on the classroom screen. Does anyone know of one? Does anyone else have refinements on the technique, or other advice for me?  If so, please leave them in the comments.

Visibly random grouping. Every single day.

Yesterday I finished the first week of my third year teaching.  I feel like I ran a marathon. (Not that I’ve ever actually run a marathon, but you know what I mean.) Each year I’ve been surprised how tired I felt at the end of the first day. Each time I have felt like I was losing my voice, but it recovers by the second day, and by the end of the week my tiredness has hit a plateau.  I seem to remember that the second week will be easier, and that I will hit my stride fairly quickly.

This year, I’m teaching two new curricula (for Algebra and Honors Geometry), and it’s only my second time teaching AP Statistics.  Add to that a switch to proficiency-based diplomas and common-core standards, and I feel like a brand new teacher. Beginner’s mind, indeed!

Given all the new challenges, maybe it wasn’t wise to throw an additional twist into the mix, but I’ve done it anyway. Each day, I’ve changed the seating in each of my classrooms, randomly assigning each student to sit at one of the six tables in my classroom.  I was afraid it would be awkward, that the students would resist, and that I wouldn’t be able to make my naturally disorganized self stick to the system. But I’m doing it, and I want to share how it’s going.

Last spring I read a paper by Peter Liljedahl called The Affordances of Using Visibly Random Groups in a Mathematics Classroom.  The upshot of the paper is that randomly grouping students every day improved group work and student interaction in a variety of ways.  The paper is worth reading.  Go ahead and click through, really!

With the exception of the AP stats class,  I let students sit where they chose on the first day. Then I explained to each class that I would move them randomly every day, and summarized the benefits suggested by the Liljedahl paper.  Each day I would deal out shuffled playing cards (aces through sixes) to the students, and students would sit at the correspondingly numbered tables.  There would be three or four students at each table, with a different arrangement every day.

Here are my impressions after the first four days.

  1. PLUS: Student seem to kind of like it.  When I described it the first day, they looked skeptical.  When I passed out cards the second day, they looked surprised that I was following through with it, but they took their cards, compared them, and took their seats.  By the end of the week, the early arrivals would ask for their cards so they could sit in the right place and not have to move later.  I was worried they would try to switch cards with each other, but no one seems to have thought  of that yet.  Or maybe they aren’t concerned enough to bother with it.  All of the students, ranging from freshmen in Algebra 1 Part 1 to seniors in AP stats, seem curious about the method, and compliant.  Maybe it’s more interesting to them than the typical seating arrangement.
  2. PLUS: So far, the classes seem more interactive than in previous years, and I’ve had few behavior problems.  There are a few “live wires” in each class, as usual, but so far they haven’t been able to settle into any particularly disruptive behavior patterns.  It may be that they are just slower to get comfortable, and that I’ll be dealing with more issues as they get to know all the other students better.  But in this first week, the classes seem much more receptive to my expectations, and they seem to be listening to each other more closely when they ask questions or volunteer answers.  They turn to look when students speak from other tables, and they direct their comments to the class as a whole, rather than just to me or to the students at their own table.
  3. PLUS: I am much less stressed about where I stand, which boards I write at, and which table I hand papers to first.  I have always worried about favoring one part of the room, so that certain students were always far from the point of instruction.  I still try to move around and use all of the whiteboards and chalkboards, but when everyone is sitting in a different place each day, it matters less. By the same token, when students go to the board to work on problems together, they don’t end up at the same board every time.  I don’t know if that really matters, but I like the idea of them getting a different perspective each day.
  4. MINUS: It’s hard to learn their names when they move every day.  I’m not very good at learning names as it is. Or faces.  I think I must have some slight disability in this area, because often I can’t remember people’s faces, even when I’ve met them a few times. It’s embarrassing introducing yourself to adults and having them say, “Yes, we’ve met before at so-and-so’s house.”  It’s also embarrassing sitting down next to a student in the lunch room who was just in your first geometry class and saying, “What math class are you taking this year?”  Anyway, it’s harder to learn names when they aren’t sitting in the same seat each day, but I think I’m learning their faces better, and I am recognizing them in the hall more quickly than I have in previous years.

I’m curious to see how this plays out throughout the year. I have up to 24 students and 6 tables, so it works well for groups of three or four. (I had 25 on my list on the first day, and I was planning to throw in a joker and make it wild, since I didn’t have room for another table.  But a student dropped, so I didn’t get to see how that would have worked.)

Have any of you tried random grouping? What is your procedure? Do you like it? Do you have any advice or things I should watch for?

Half-similar papers update, or I’m not as awesome as I thought.

OK, I came up with an investigation in my class that I thought was really awesome. (See previous post.) It was simple, engaging, and collaborative, and the students actually argued about it and then rolled up their figurative sleeves and worked it out with only occasional intervention from me.  In addition, there were multiple ways to solve it, and there was room for extension at the end.

But I felt slightly less cool when I realized that the next section of the textbook had a homework problem that was almost exactly like it. (It was couched in more pseudo-context about a greeting card manufacturer who for some unfathomable reason cared whether their card, when folded, was similar to the unfolded state….)  So I’m not as blindingly original as I felt in the moment.

Nevertheless, I assigned the problem as part of the next night’s homework assignment.  That’s when the real ego smack-down happened.

They had no idea how to solve it.  I don’t think a single student could do it. (At least, no one would admit to solving it, which is another problem in classroom culture I need to deal with.)  This was almost EXACTLY the problem they did as groups THE DAY BEFORE.

Here’s me.

So what is the deal?  Does group work feel good to us as teachers, while not really teaching the students anything?  Does it make us think they are learning individually, when really they are just leveraging each other’s partial understanding?

What can I do as part of group work to help each student to consolidate the whole process?  (I am thinking journalling about the problem, but that’s something for next year, because there’s not time in the schedule to establish that as a classroom habit at this point.)  Is there something not as time consuming I can do (or have them do) as a wrap-up?

Help me, math twitter blogosphere. You are my only hope.

Half-similar papers: a PBL in similar polygons

I got the first inklings of an idea for this lesson months ago when I tried to photocopy a two-up layout (two letter-sized portrait pages onto one letter-sized landscape page) using a 50% reduction. Doh! I wasted a piece of paper, but it got me thinking about scaling. Against all odds, I remembered it this weekend when planning my similarity unit.

We are beginning discussion of similar figures in my honors geometry course. The textbook introduces similarity with transformations, in a way that I feel is quite non-intuitive, even though we’ve been using transformations quite comfortably throughout the course to define and discuss congruence. I think the students understand thoroughly that dilations are a size change. We dilated our favorite cartoon characters, and we got some very interesting results. (More on that in a later post.)

But the book first defines similarity in terms of “a composite of a size change and any number of reflections.” That idea requires so much unpacking and consideration that I feel it belongs later in the chapter, after the students get a feel for what similarity looks like. The section immediately after the definition discusses the relationships between perimeters, areas, and volumes of similar figures. So I glossed over the transformational discussion (to come back to it in a few days), and posed two very simple questions to get them thinking about how similarity works.

The half-similar paper question:  I held up two letter-sized (8.5 by 11 inches) pieces of paper.  I folded one in half horizontally and held it up next to the other piece. (I didn’t have my camera at the time, but this is what it looked like:)


“Are these shapes similar?” I asked. There were a flurry of immediate answers.


“No, both sides aren’t half!”

“No, two of the sides have the same length!”

I thought about it for a second and asked, “So shapes can never be similar if one pair of sides is the same length, but the other isn’t?” (I wasn’t sure what they were getting at, but I didn’t want to wreck the vehemence with which they were arguing.)  Some of the students looked confused, and some tentatively said, “Yeah….”   So I looked around the room and found a pad of square post-it notes.  I folded one of them in half and held it up next to the whole one:


“Does this look the same as the first case?” I asked.  The students immediately noted that these two shapes looked “even less similar” than the first set.  So I asked, “Do you think it might depend on the original shape of the paper?” They looked thoughtful, and I think some of them saw the point: if this case is worse than the first, then another might be better….

I let them wrestle with the problem in groups. I saw much measuring (yes, if you fold paper in half, it’s half as wide) and several approaches using proportions, and every group managed to come to the conclusion that the two shapes were not similar.   When each group reached that conclusion and explained it to me, I set that group loose on the follow-up problem: what shape of paper WOULD work? I made it simpler (maybe too simple, I don’t know) by saying, “The long side is 10 inches long. How long does the short side have to be if the folded paper is similar to the unfolded paper?”  Eventually (although some groups needed more guidance here) they all came up with sqrt(50).

(By the way, this led to an excellent discussion of simplifying square roots, and another justification for teaching that skill in the first place. Unless they can simplify the square root, it’s not at all obvious what the proportional relationship is, and how to scale to other sizes of paper.)

Overall, I felt like this went really well.  I do wish I had spent more time trying to get them to generalize. I originally thought of this as an area problem ( the area is divided by two, but the side length is divided by sqrt(2)), but we never really got there. Instead we moved on to a volume question (which I meant to post about later, but now I don’t remember what it was–Dang).  But I think this helped the kids to wrap their brains around what it really means to ask, “Are these similar?”

Does anyone have suggestions for making it better?

Postscript: When I got home, I did some research, and I came up with an interesting follow-up question, maybe for a few days later: “Why is A4 a more sensible paper size than US Letter?” Besides the metric system, of course…

Post-postcript: Well, wipe that smug smile off my face.  Thanks to assessment the next day, I found out that most of the students didn’t actually learn much.  Are there any ideas for an end-of-class routine that would help the students to individually consolidate the group learning?

The how of standardized tests

Yesterday over donuts and coffee, I had the beginnings of an interesting conversation with a colleague. Our juniors are scheduled to take the Smarter Balanced Assessments (SBAC) at the end of this week, and it has been the source of a great deal of controversy over the past few months. MANY of the students are opting out, our federal funding is still officially at stake, there is a flurry of legislation pending in the state legislature, and there are very strong opinions on all sides.

I say it was the beginnings of an interesting conversation, precisely because of those strong opinions.  Since our school is required to give the tests, and since the students not opting out of the test will invest significant energy in doing well, I feel strongly that we should do our best as a school to get real, usable data.  To do otherwise means our results are essentially useless to us, even as a baseline for future testing.

My colleague feels (just as strongly) that the SBAC assessments are (1) damaging to kids because they test things the kids have not been taught in our current curriculum, and (2) part of an unethical, profit-driven cycle that we should disrupt in every legal way possible, including by opting our own children out of the testing and encouraging others to do the same.

Strong opinions can lead to strong reactions.  But as my neighbor the Zen priest says, “Just because you see the bus going by, that doesn’t mean you have to jump on and ride it wherever it is going.” In other words, don’t react without thinking.  Listen first, then consider, and then respond. Both my colleague and I did a good job on the surface; we appeared to listen thoughtfully to each other and discuss our disagreements like grownups.  (At least my colleague did, and I hope I did, too.)

But on my side, the key phrase is “appeared to,” and I know that part of that appearance was an act. Granted, I wasn’t just responding by reflex. I was thinking carefully before I spoke, largely because I really respect my colleague and didn’t want to offend her.  But mostly I was thinking, “How can I express myself really clearly so it’s obvious that I am correct on this?”  I was granting the validity of her arguments, but not really trying to put myself in the position of understanding and supporting her opinions.  I know that it’s intellectually lazy of me to disagree with someones opinions unless I’ve made that effort.  It also deprives me of some really interesting ideas.

So, with apologies to my colleague for not doing this in the moment, here I go.

Argument one against SBAC testing: It damages children to test them on content they don’t know.

I do believe that many, maybe most of our students are upset when they face test questions they can’t immediately see how to answer. Some of them are demoralized; they see it as confirmation that they aren’t good at the subject. Others get angry that they are being unfairly assessed on content they haven’t been taught.  But whose fault is that?  I think it is something we as educators have brought on ourselves, by never presenting kids with test questions that they shouldn’t immediately know how to answer. We scaffold the heck out of everything.  We try  to teach each student in Vygotsky’s zone of proximal development, but we take great care never to test them there.  We train students that we hope for them to get every answer correct, and we are disappointed when they don’t. We make fixed grading scales that penalize the students when the teacher writes a question they can’t wrap their heads around yet.  As a result, neither the teachers nor the students ever really find out what they can do.

This is poor preparation both for college and for life.  Both places are chock-a-block full of questions we don’t know how to approach.  Often our half-formed answers and attempts at answers have really important consequences.  We should be preparing for our entire lives to climb all sorts of challenging cliffs, hoping to find a hand- or toe-hold that will let us get up to the next ledge.  It should be fun and invigorating to do this.  As teachers, we need to prepare our kids in such a way that the majority of them get the chance to respond to challenges in this way, rubbing their hands together in anticipation. So how do we go about it?

My colleague is right.  We don’t do it by throwing a test like this at them for the first time when they are juniors in high school.  We don’t do it by throwing it at them once a year at “standardized testing time.”  But I’m right, too.  We really need to do it somehow, and we need a measure that shows we aren’t doing it now.

Argument two against SBAC testing: it is part of an unethical, profit-driven cycle that we should disrupt in every legal way possible.

I definitely agree that standardized testing has become way too profit-driven, and that a whole exploitative industry has grown up around our quest for data about our students.  I’m not sure how we should respond as teachers. I think it is really important not only to assess our students on what we have taught them, but also to know how their learning (and our teaching) compares to that in the rest of the world.  Any attempt to measure such a broad question is going to be imperfect, but that doesn’t mean the measurement isn’t important.

A bigger question is how to keep profit and exploitation out of the equation.  Is it even possible in our system? In an education system where every state and sometimes every local school district has a different curriculum, whom can we trust to come up with standardized measurement tools? How can we all have input on what it should look like?  How can we both use the data as educators and keep the data from being mis-used by others?

Any thoughts? More questions? Please leave them in the comments!