I'm My Own Best Teacher

I had one of those nights last night that I truly enjoy, but don't often reflect upon. It all started when I was going to write a post about something Lorrie Shepard said at the Realistic Math Education Conference about instructional and assessment tasks. But first, I thought, I should look to see what she's written about the subject so I would have a greater background.

Since I'm not currently at the university, my access to full research articles is restricted, but I know CU has a VPN through which I can look like I am at the University. I found the VPN instructions on CU's website, but because I run linux the setup was more complicated. To their credit, the university had a linux client you could download and instructions for installing it, but I wanted to use whatever client that was already in the Ubuntu repositories. It took a little while, but after downloading the university client and sorting through the source code and configuration files, I had the information I needed to get the native VPN clients running on both my netbook and workstation.

Now, back to the Shepard article. Now that I had the same access I'd have if I were at CU, I could download full copies of papers from multiple sources. I was using Google Scholar, and lately I've been trying out Zotero to manage bibliographic sources. Instead of focusing on the papers, I was focusing on how the bibliographic entries were being stored by Zotero. Zotero wasn't detecting the sources very well on its own, but I was having good luck choosing Scholar's "Import into BibTeX" option,copying the entry to the clipboard, then telling Zotero to import from the clipboard.

I'm a long-time LaTeX user, but I've never used BibTeX. For a while now I've been wanting to reformat my undergrad senior thesis I wrote in 1999, but to do so I knew I'd want to re-manage the sources I used. I found my old thesis and started searching Google Scholar for the same references and I stored them in Zotero using BibTeX imports. I had imported about half the references when I could not hold back my curiosity any longer. Could I export these from Zotero and use BibTeX and LaTeX to re-format my paper?

After a wild goose chase trying to get the Gentium font in LaTeX (I guess I'll have to make that work some other day), I spent an unknown amount of time learning how BibTeX worked with LaTeX. My thesis had been formatted in some unknown format before (something for which I had always been disappointed in myself), so this time I was determined to format the paper using an APA style. There is an APA style package for LaTeX, and after much troubleshooting I was on my way to formatting the thesis. I went to bed at 5 am with the second half of the paper left to reformat.

What's the point of telling this story? Look how much I learned through my curiosities and distractions! I learned how to configure a VPN on linux, how to import, edit, and export bibliographic entries in Zotero, how to use BibTeX, and how to get LaTeX to automatically format papers in the APA format. I had been tired throughout the day, but learning all these things drove me well into the next day, hardly tired at all.

Everybody seems to have an answer to the question, "Who was the best teacher you ever had?" My answer? Me. I'm my own best teacher. I don't mean to sound cocky; in fact, I can only make such a claim because I'm humbled by the effect that education, including my many teachers, has had on my life. It's because of them and the many great opportunities they provided that I have become the life-long learner they wanted me to be. Teachers gave me the knowledge and skills I would need to learn on my own and become my own best teacher, and for that I'm forever grateful.

Cases in the Ethics of Grading: Jodi Warren and Mr. Kennedy

The following is my fourth and final (as far as I know) in a series of hypothetical cases meant to raise questions about grading practices. I'd like to recognize Kenneth Strike and Jonas Soltis for their book "The Ethics of Teaching," which inspired the style and structure of this case. Enjoy and discuss!

Louis Kennedy and Jodi Warren are two teachers within the Metro City School District. Louis is a popular fifth grade teacher at Worthington Elementary and Jodi is a new, 23-year-old sixth grade English teacher at Carter Middle School. Most of the students at Worthington Elementary matriculate to Carter Middle, and many of Jodi's students had Mr. Kennedy as their fifth grade teacher.

At her very first day at the school, Jodi Warren listened to her principal preach some of the reforms they were instituting that year at Carter Middle. The biggest change was the enforcement of course requirements and credits that must be earned before being moved on to high school. "No more social promotion!" the principal cried. "We will teach these kids to be responsible and prepare them for high school!" Jodi wanted to be tough her first year of teaching, so this was a philosophy she was willing to get behind, even at the sixth grade level. She'd heard horror stories of teachers who were pushovers and let the kids run the classroom, and she was determined to not be one of those teachers.

Several months passed and Jodi stuck to her plans of holding kids responsible and grading rigorously but fairly. Many students were earning grades of C or lower, but Jodi did not seem too concerned, as she had always viewed a C as an average grade. At the end of October, she faced her first parent teacher conferences. Parents were not happy. Conference after conference, Jodi listened to parents make comments like, "This is the worst grade my student has ever gotten," and "My son was doing much better in Mr. Kennedy's class last year." Jodi tried to explain the new expectations at Carter Middle, and how it was part of a plan to better prepare students for the rigor of high school. "Well, my daughter has As and Bs in all of her other classes," some would reply. Jodi also learned that a number of the parents had already complained to the principal about her ineffectiveness as a teacher, and it did not appear that her principal gave her much support. Jodi left school that night with serious doubts about her future as a teacher, wondering if her teaching really had been so poor despite her best efforts.

The next day Jodi attended a district workshop where she had an opportunity to work with other teachers, including Louis Kennedy. Jodi was curious to hear if Louis had experienced problems similar to those she was having with students Louis had taught the year before. Maybe she could gain some insight as to what she was doing wrong, or pick up advice from a popular fellow teacher. When Jodi asked about trying to be a tough grader, Louis gave her a surprising reply. "We give A-F grades in fifth grade, but I never take it too seriously. They don't count for anything. They don't end up on a transcript for college or factor into a GPA. A few years ago I had given a couple students Ds and Fs, and I found out that it only created animosity between myself and the students. They weren't happy in class anymore and parents assumed I was doing a bad job. Now I never give a grade below a C, everyone seems to be happier, and I have an easier job connecting with struggling students."

Jodi is now unsure what she should do. If she suddenly raises her students' grades, she'll become the easy pushover she did not want to be. If she keeps grading the way she is, parents will think she is not a good teacher and that could put her future as a teacher at risk. If she complains about Mr. Kennedy's easy grading practices she'll be seen as whiny and a tattle-tale, a trait she despises in her students.

  1. Does it make a difference that Jodi is a new teacher, and has several years of teaching before she receives tenure?
  2. Given Jodi's choices (raise grades and be a pushover, be firm on grades and risk her career, or complain about Mr. Kennedy) which would you choose and why?

Cases in the Ethics of Grading: Mrs. Lemon and Troy Mann

The following is my third in a series of hypothetical cases meant to raise questions about grading practices. I'd like to recognize Kenneth Strike and Jonas Soltis for their book "The Ethics of Teaching," which inspired the style and structure of this case. Enjoy and discuss!

Troy Mann is a struggling student in Mrs. Lemon's Algebra 2 class. By all accounts from Troy's previous two math teachers, he barely passed Algebra 1 and Geometry, and his Geometry grade might have been artificially inflated to compensate for his weak skills coming out of Algebra 1 and suspicions of a possible undiagnosed learning difficulty. Still, in both classes, Troy's modus operandi was the same: "coast" through as much of the semester as possible, do as little work as possible, and then apply just enough sincere effort to avoid being ineligible for sports. Troy understood if he failed Mrs. Lemon's Algebra 2 class he would be ineligible for the beginning of the second semester, meaning he would miss most of basketball season. Troy expected to start on the varsity team, and Mrs. Lemon happened to be one of the biggest basketball boosters at the school.

Using athletic participation as a motivator, Mrs. Lemon and Troy began to meet almost every day after school to improve his math grade. Unfortunately for Troy, his lack of effort earlier in the semester and in previous classes left him unprepared for the rigors of Algebra 2, and most of his time with Mrs. Lemon was spent relearning skills from Algebra 1. After six weeks Troy was showing great improvement on his Algebra 1 skills, but was running out of time in the semester to master Algebra 2 skills well enough for the final exam. Unable to fully catch up, Troy badly failed the final exam and received an F for the course.

When school resumed after winter vacation, Mrs. Lemon was called to the principal's office to meet with Troy's parents. This was the first time Mrs. Lemon had met Troy's father, but had talked previously on several occasions with Troy's mother, who worked in the school district administration office. The principal explained to everyone that a failing grade would mean Troy would be ineligible for basketball. Both of Troy's parents emphasized how important sports are to Troy, and without sports he may not have any motivation to be successful in school. Troy's parents also expressed their difficulty in understanding how Troy could spend all those hours after school, learning and showing steady improvement, and still fail the course.

Mrs. Lemon explained that to pass the course students must show mastery of Algebra 2 objectives, and while Troy should be commended for remediation of his weak math skills, it was not worthy of Algebra 2 credit. When the principal asked if there was any way of making Troy eligible for the second semester (implying Mrs. Lemon should change the grade to passing), Mrs. Lemon bluntly suggested that he ask the athletic director to ease the eligibility standards instead of compromising her own academic credibility. With that, the meeting ended and Troy's parents left.

Before letting Mrs. Lemon go back to class, the principal stopped her and said, "I want to tell you a little story. When I was in college, I really struggled in some of my classes and for one class I failed the final and was going to fail the class. The professor for that class, instead of sticking me with that grade, brought me over to his house, made me dinner, and went over the test item-by-item. I didn't retake the test – he was just helping me understand the material. He changed my final exam grade and I passed the class. Think about sitting down with Troy and doing him the same favor."

  1. State content standards are meant to guide curricula, but should mastery of that content be the sole determinant of a student's grades? Should Troy's grade reflect his effort and skill improvement?
  2. Was it ethical for the principal to ask Mrs. Lemon to consider changing Troy's grade in the presence of Troy's parents? What's more important, the perception of a principal's fairness and neutrality or his/her willingness to be open and helpful to students and parents?
  3. What do you think was the principal's intent of telling the personal story after the meeting?
  4. If you were Mrs. Lemon, would you change Troy's grade?

Cases in the Ethics of Grading: Mr. Green and Tracked Classes

The following is my second in a series of hypothetical cases meant to raise questions about grading practices. I'd like to recognize Kenneth Strike and Jonas Soltis for their book "The Ethics of Teaching," which inspired the style and structure of this case. Enjoy and discuss!

Mr. Green is the sole history teacher at a small high school. Last year the school experimented with having an honors science class and feedback from teachers, parents, and students was generally positive. This year the school has decided to expand its selection of honors courses and Mr. Green will be responsible for teaching the school's first section of Honors World History. The class is designed for 10th and 11th graders, almost all of whom Mr. Green has already taught in lower-level courses.

Because of the school's size, there are only enough students to warrant two sections of World History: one honors and one regular. Mr. Green is responsible for selecting which students are to be placed in the honors section, and he overwhelmingly and sensibly chooses students who have earned grades of A or B in his previous courses. That naturally leaves almost entirely C or below students for the non-honors section of World History. Mr. Green is fully aware that the creation of Honors World History amounts to tracking, a controversial subject thought by some to be outdated and unethical. To ease his own concerns, Mr. Green ensures that both classes receive the same curriculum, the same textbooks and materials, and the same styles of instruction. To make the honors section worthy of its title, he skips the chapter review day before each test, thereby slightly speeding up the class, and makes the grading scale marginally more challenging.

The first semester of the two sections of World History seems to go smoothly. Mr. Green's plan to teach both classes using the same materials and methods appears to have avoided controversy. Having not heard any complaints from students, parents, or administrators, he proceeds into the second semester using the same class policies and procedures.

Three weeks into the second semester, the Mr. Green's principal, Mary Williams, checks the grades for both honors and regular World History. She is upset to find that while there are plenty of students earning As and Bs in the honors class, not a single student in the regular World History class has a grade above a C. Ms. Williams calls Mr. Green into her office, accuses him of grading unfairly, and demands to know why the regular World History class doesn't have "its own As and Bs." Mr. Green offers several reasons, including: A) Tests usually help students raise their grades, but it's early in the semester and they haven't taken a test yet, B) The regular World History class doesn't have students with a previous record of earning As and Bs, so the lower grades should be expected; and C) It would be unfair to arbitrarily give As and Bs to students in regular World History if they weren't mastering the content at levels similar to the honors students, and doing so would take away the incentive for students to take Honors World History.

Ms. Williams is not swayed and flatly tells him that he "needs to give higher grades." When Mr. Green asks, "Does that mean I should give higher grades without regard to ability or achievement?" the she only responds by repeating herself: "You need to give higher grades."

  1. Is it reasonable for Ms. Williams to expect each class to have a distribution of grades A through F?
  2. Suppose you raised some of the grades in the regular World History class, believing the class does deserve a broader distribution of grades. Would it be hypocritical to do this unless you also lowered some of the grades in Honors World History to include more Ds and Fs?
  3. Did Mr. Green trap himself in this dilemma by trying to make the two sections so similar? In other words, do you think he would have been more comfortable assigning each class "its own As and Bs" if the two classes were drastically different by design?
  4. Schools use grades and GPA to measure and sort students. For honors vs. regular scenarios such as this, whose responsibility is it to ensure the sorting is done fairly? Is it Mr. Green's duty to sort the students across both sections, as he was attempting to do, or should the district handle that burden through policy? (Example: Districts often choose to weight honors courses on a 5-point scale to reflect their increased difficulty over regular courses.)
  5. If you were Mr. Green, would you raise the grades? If so, why? If not, why not?

Cases in the Ethics of Grading: Dr. Jones and Tara Hightower

The following is a hypothetical case meant to raise questions about grading practices. I'd like to recognize Kenneth Strike and Jonas Soltis for their book "The Ethics of Teaching," which inspired the style and structure of this case. Enjoy and discuss!

Dr. Susan Jones, an assistant professor in her first year at Central State University, is teaching a molecular biology course to a class of about twenty undergraduates, most in their second or third year of college. The content of the class comes easy to Dr. Jones, but showing up on-time at 8:00 three days a week is not. Most of the class is struggling with the material and disliking the class, so to make the class more enjoyable she occasionally interrupts lecture with some "fun" activities, such as watching cartoons or playing games. Dr. Jones's primary suggestion to those struggling in the class is for them to see her during her office hours for individualized help. She is very generous with her time and has helped many students make improved progress with the coursework.

Tara Hightower is an honors student at Central State and is one of Dr. Jones's struggling students. Tara is proud and independent, and somewhat stubbornly chooses to not see Dr. Jones for individualized help. Tara attends every class, is always on time, and studies both the text and her notes for hours each week in order to keep up with the material. This has always been a successful strategy for Tara in the past, and she resents the idea that she should have to see Dr. Jones individually, especially since some class time is already being wasted on fun, games, and Dr. Jones's tardiness. Tara's scholarship and status in the honors program is dependent on her maintaining a 3.5 GPA, and she received a notice at midterm that she had a D in the class. If Tara can not substantially improve this grade, she'll be given a warning by the honors program and risks losing her scholarship.

At the end of the semester, Dr. Jones asks Tara to come see her about her final exam and grade for the course. At the meeting, Dr. Jones explains to Tara that while she passed both the homework and the final exam, she did not perform up to expectations and should take the course again. To ensure Tara retakes the course, Dr. Jones assigns her a failing grade. Tara feels that receiving an F for passing work isn't fair, but agrees that her performance was sub-par and knows she needs to retake the course, regardless if her grade was a D or an F. After meeting with her advisor, Tara changed her schedule so she could retake the course with a different professor the next semester. Tara received an A on the retaken course and regained a positive standing in the honors program.

  1. Can the extra hours Dr. Jones spends working with students individually make up for misused class time?
  2. Should Tara's refusal of any out-of-class individual help influence the grade she receives from Dr. Jones?
  3. Should the fact that Tara must retake the class, regardless of earning a D or F, matter to Dr. Jones?
  4. Teachers typically retain autonomy over their own grading practices. If Tara were to choose to protest the failing grade, who should have the power to change it? What are the ethical implications for Dr. Jones if her grades can be overturned?

Grading and Teacher Autonomy

I've known some teachers that probably wish they had the freedoms teachers enjoyed in the one-room schoolhouse days, but in the modern context of standards-based education that just isn't a reality. Standards, for better or worse, shape our curriculum, choice of texts, long and short-term unit/lesson planning, instruction, and seemingly every level of assessment. Could there possibly be a significant daily teacher practice not guided by standards?

Sure there is: grading.

There are no national or state standards to tell you what a "B" means, no standards to tell you if you should use a weighted grading system, and no standards to tell you if habitual tardiness to class should result in a grading penalty. I know of no set of classroom grading standards published by any major educational organization at any level. Teachers are left to figure out grading for themselves. As a math teacher I put extra pressure on myself to develop the best possible grading system, and over six years I tried all sorts of variations, none of them perfect.

My general philosophy was to grade on the mastery of the content described by the standards, and leave out as much extraneous stuff as possible. (Like deducting points for tardiness, for example.) Even on that task I know I failed, because the vast majority of student homework was graded for completion, not correctness. Still, I felt I had a principle for grading and I tried to stick to it. I actively resisted meaningless grade inflation and expected grades to usefully measure what students knew and could do. I felt it was reasonable in Colorado, where more than 60% of high school students score partially proficient or below on the CSAP, for Cs and lower scores to be an acceptable reality. Also, I hoped my students' grades would have a strong positive correlation to their CSAP scores. (Sadly, I never tested this.) Grades that failed to correlate would have indicated problems with my grading system and been a disservice to my students. I was miles away from having this vision be true for all students all the time, but it was a goal nonetheless.

I hope you can get a feeling for the amount of thought I invested in my grading, and realize that teachers all do this to some degree as a result of the autonomy we have over grading. The autonomy isn't total, however. If I had given 95% of my students As and Bs, I probably wouldn't have had much reason to continually re-examine my grading practices. In most schools, a teacher who gives that many high scores will certainly avoid negative attention, or perhaps even be praised for being such a good teacher. Low grades, on the other hand, attract plenty of attention from students, parents, and administrators. You can get many interesting suggestions from all involved, including grade curving (which means raising in this context, trust me), extra credit, and throwing out low test scores. Most of this is based on appeasing students and parents, not the development of intelligence, but it happens in schools large and small.

So, with no grading standards, to whom does a teacher surrender their grading autonomy and to what degree? The ethical quandaries can build extremely quickly, and several such cases will be topics here on this site in the near future. One of those cases will present a situation where some might argue the teacher got total autonomy and used it to be purposely unfair to a student. Another will present a teacher being pressured to change a grade for non-academic reasons. Hopefully these cases will challenge you to think about your own classroom, and good reasons to either defend or change your own grading practices.

Lastly, I want to pose a question about a possible influence on our grading systems. This is not a who, but a what: How does your gradebook itself influence your grading? Most schools use a system like Powerschool or Infinite Campus. (I've used Goedustar and Integrade Pro.) How do the capabilities of the program, including calculation methods and input limitations, affect the way you grade? Do you find technical limitations an acceptable influence on your grading practices? Also, how do you feel about parents essentially having real-time, 24/7 access to their child's grades? Is this an unreasonable or unhealthy demand? I'd love to hear your thoughts!

FIUS 2009: Curious Minds

Wednesday Morning Keynote: "Curious Minds: in search of scientific reasoning skills in pre-school children"
Speaker: Jan de Lange, Freudenthal Institute USA

Jan started out as a mathematician and when he joined the Freudenthal Institute his interests were mainly in upper-secondary education. Gradually, he worked down the grade levels to lower elementary, and, combined with his experience of becoming a father late in life, that sparked a great interest in the reasoning skills of pre-school children.

Research suggests that the curious minds of young children are underused; both formally and informally, more could be done to maximize their cognitive capacities. It is the mission of Curious Minds (PDF) to chart the talents of young children and to investigate how they can be kept alive, especially in the area of scientific reasoning and problem solving. The Curious Minds project brought together developmental psychologists, neuroscientists, behavioral scientists, and mathematics education experts. Previously, these groups had never worked together.

The research questions are:
  • How do the talents differ among tasks and children?
  • How do they develop in individual children?
  • Are the 'talents' observable in the actions and evaluations of children in talent-eliciting tasks?
  • What is the role and development of language?
  • What role play contextual factors?
  • What is the predicitive value of talent at a young age?
  • How do kids reason?
This is an intensive project that will follow 20 students for 20 years. They have no plan (but do have funding) as they are in search for something and need as much freedom as possible.

"We are born with navigation and spatial reasonings skills, but we do little to develop it. So 20 years later we need a Garmin or a TomTom."

The immediate focus for the Curious Minds project is to build a network of researchers, map the talents and effects of their interventions, develop talents, identify variables that influence talent development, and identify talent eliciting tasks and useful materials. From 2011 onwards (the project is funded through 2017) they are looking to collect data through experiments at daycare centers and primary schools, performing pilot studies about the role of parents, and development of new tasks.

In Jan's opinion, the biggest threat to the project is that everybody wants to get on the bandwagon, and people's natural impatience will lead them to make immature conclusions and improper implementations. In any case, says Jan, playing with the kids "will be the best way to spend my retirement."

(Sorry I can't post the videos shown of kids interacting in the activities. They're fascinating!) [Edit: if you can make your way through the Dutch (at least I think it's Dutch), the videos are online at the project website.]

FIUS 2009: Realistic Math Education Conference

One of the advantages of leaving the classroom and returning as a student full-time is that you can participate in conferences and meet a wider circle of people who introduce you to resources you previously did not know to exist. This week I'm at the Realistic Mathematics Education Conference, presented by the Freudenthal Institute USA. FIUS is located here at the University of Colorado at Boulder and its director is Dr. David Webb.

The Freudenthal Institute was established at Utrecht University in the Netherlands by Hans Freudenthal in 1971. Freudenthal believed that math is a "human activity" and that students would learn it best if math was not presented as a static, discovered, formulated subject. Instead, the Freudenthal Institute adopts theories such as "progressive formalization," representing an informal to pre-formal to formal progression that students experience as they view mathematical situations in real-life contexts. The Freudenthal Institute is particularly strong in design theory, and Freudenthal Institute staff have contributed to research, textbooks, and international tests, always with the goal of grounding math in the context of real-life.

Today is the last day of the conference and I'll try to summarize some of the particular talks this afternoon. There are people here from 22 states and all over the world, including the Netherlands, Japan, the UK, and Nigeria. The work people have been presenting is very solid and I must admit that in my case I feel like they are preaching to the choir. The math wars aren't over yet, though, and those who fight are probably the teachers who don't even know the wars exist. Professional development like the FIUS/RME conference provides opportunity for further reform, but reform is never quick or easy. For example, check out this video clip:

"Grades...Are a Fantasy"

Chances are, if you're a teacher, you are solely responsible for assigning grades to your students. Rarely is it a pleasant process. You may have taken an evaluation and measurement class as part of your teacher preparation program, but now you're in a real classroom, with real students (who have real parents) and you get to become each child's judge, jury, and in some cases, executioner. It's not one of the easier parts of teaching, because we as teachers pressure ourselves to make our grading as accurate and fair as possible.

But what is "fair"? Just as there is no perfect or even universally-agreed definition of the purpose of education, there is no common understanding on exactly what grades do and what a grade means. Every semester, just when I think I have my grading system figured out, I'll notice students with lower grades that I consider to have mastered content at higher level than some of their higher-scoring classmates. Do I let my professional judgment trump my grading system and change the grade? Why not? After all, I was the one who designed the grading scale in the first place. The grading system is flawed either way.

If grades can create such a quandary for the teacher, imagine the myriad of possible meanings they have for students. Some think grades are earned for simply turning in work. Others think test scores are the only critical factor. Still other students depend on grades to reflect their effort, absent any measured achievement. With so many interpretations, it's a wonder grades hold any meaning at all. I'll never forget the words of my mentor Bob Anderson at Florence High School. "Grades...are a fantasy." He was implying that students often knew so little about what their grade represented and what it took to raise it (especially at the end of a grading period) that grades might as well not be based in any reality at all. I always got this feeling when a student would ask, "If I do well on the test, will it raise my grade?" Not only did they not understand grading, but apparently they didn't understand how averages worked, either.

I've given grading a great deal of thought, probably due to pressure I put on myself as a mathematician to develop a perfectly fair, objective, and accurate grading system, and failing every single time. Consider this post as the first in a series dealing with multiple aspects of grading such as teacher autonomy, influences of grading software, and ethical questions presented from case studies.

As a final thought, I realize my transition from teacher back to student means I'm again receiving grades instead of assigning them. When I got my first graded paper back in my Nature of Mathematics Education class I looked at the letter grade like it was some sort of novelty. In fact, I was surprised it was there at all, and I really didn't care what it was. (Okay, maybe I would have cared if it had been lower.) What I really cared about was the comments left by my professor, and any hints on what I could have explained better. Apparently I gave up on the grading fantasy a long time ago, and I think I'll sleep better knowing so.

Beyond the Blacklist

Internet filtering is a hot topic, highlighted by many bloggers recently during Banned Books Week. At my school last year, our tech staff implemented their solution to filtering: a whitelist. (I should be hearing audible gasps from across the internet as you read this.) Instead of blocking potentially harmful sites (which included anything "distracting" at this school), they argued it would save everybody time and trouble to just allow kids to use a hand-picked version of the web, and make it easy for teachers to request sites they wanted added to the whitelist.

To their credit, our tech staff did not make this change lightly or without involving the staff in the discussion. They assured us it would be okay: all *.gov and *.org TLDs would be on the whitelist (I'm not sure if they knew that anybody could get a *.org, no different than *.com), they were going to promote use of the Librarians' Internet Index, and new sites could be added by teacher request. Also, teachers' computers had totally unfiltered access to the web, so we wouldn't be inconvenienced by rules meant for students (which reduces teacher complaints/awareness). For students there would be no Facebook, no YouTube, and no Google. (The "no Google" policy didn't last long - the school provided a custom Google search several weeks after the whitelist was implemented after many complained. It still only searches the whitelist, however.)

The whitelist survived the rest of the school year and remains in place. It doesn't help that few staff members are what I would consider web2.0-savvy. (Some of our teachers had to be taught a couple years ago that spreadsheets could be scrolled left and right, not just up and down. Asking some to teach their classes how to edit Wikipedia would be like asking them to take their class on a field trip to the moon.) With predominantly tech-novice teachers, the whitelist will remain, available resources will be underutilized, and the information gap will grow. Students aren't allowed to use their mobile phones and the tech staff practically lives in fear that a student will bring a laptop into the building and want internet access. I'd like to believe this school is an exception to the norm, but it isn't. Restrictive access to information and technology tools leaves students and teachers to search for work-arounds, such as described by Dr. Alec Couros in his post, "Freedom Sticks For The Classroom."

We spend a lot of time trying to ensure our content is relevant to our students, but increasingly how we deliver content is what is losing relevance. Students understand how technology increases their power and access socially, and expect technology to increase their power and access educationally, too. Schools need to rethink their policies, such as described by Will Richardson in his post, "Don't, Don't, Don't vs. Do, Do, Do." So with less restrictive access, how do we encourage effective and productive uses of technology? We teach. We teach students how to search, how to judge the quality of information, how to avoid distraction, and how to give back to that great body of knowledge that is the internet.

To quote Ira Socol, "let's follow up 'Banned Books Week' with 'Banned Sites Year' - a commitment to replacing filtering with education and intelligent conversation." Get your tech staffs involved, and hope they adopt stances like St. Vrain Valley here in Colorado, as described by Bud Hunt in his post, "Would You Please Block?":
What we’ve decided is that we will no longer use the web filter as a classroom management tool. Blocking one distraction doesn’t solve the problem of students off task – it just encourages them to find another site to distract them. Students off task is not a technology problem – it’s a behavior problem. It is our intention that we help students to learn the appropriate on-task behaviors instead of assuming that we can use filters to manage student use. Rather than blocking sites on an ad hoc basis, we will instead be working with folks to help them through computer and lab management issues in a way that promotes student responsibility. We know that the best filters in a classroom or lab are the people in that lab – both the educational staff monitoring student computer use as well as the students themselves.
The internet is the greatest information resource in the history of the world, by far. Access to that resource is more abundant than ever, and people (including students) expect access. (Most students carry a device with them that gives them access, and we tell them to put those devices away.) If we aren't willing to move beyond the blacklist, we need to seriously reconsider what we believe the purpose of education to be. That's a big debate for another day, but I don't think any of us would agree education should be about the restriction of access.

CCTM 2009: An Overview of the New Standards/Revisions to the High School Math Standards

Presenter: Melissa Colsman, Colorado Department of Education
Presenter: Julie Stremel, Aurora Public Schools

I attended two sessions regarding the revisions to the Colorado math standards, the first covering a broad overview of the revision process for K-12 and the second looking more specifically at the changes being made to the high school math standards. The first session was presented by Melissa Colsman, math content specialist from CDE, and the second by Julie Stremel of Aurora Public Schools. Stremel is a member of the standards review committee.

Standards revision in Colorado is being driven by two pieces of legislation: Senate Bill 212 (CAP4K) and House Bill 1168. The standards are still in the revision process, but barring any major changes or deliberation they should be ready to submit as a final draft to the state board of education within the next several months. The goal is to create standards that are "fewer, clearer, and higher." We will be shrinking from six standards to four, although it's not clear that any significant content is being removed rather than reorganized. (Example: Previously geometry and measurement were two standards and now they have been combined into one. I'll leave it up to you to decide if that qualifies as "fewer.") The new standards have been developed with a single goal in mind: producing competent, prepared high school graduates who are ready to succeed at 2- and 4-year colleges, trade schools, the military, and the workforce.

As for clearer, CDE is removing the repetitive language from grade to grade by only listing a standard at the grade level it is expected to be mastered by students. CDE is letting districts, schools, and teachers decide how to build up to that mastery and for how long. For example, a particular algebra standard might be listed for 7th grade, but students will need to be introduced to that topic in 5th grade and take three years to build to mastery level. The current standards made a greater effort to describe the progression towards mastery from grade to grade, but too often the distinctions between grade levels were too vague to be helpful.

Perhaps the biggest change to the math standards comes from House Bill 1168. That bill called for greater financial literacy for students and CDE has decided that those parts of personal financial literacy (PFL) that are mathematical in nature will be assessed on the math CSAP. Schools are going to have to determine who is responsible for teaching that material, whether it be a math teacher, business teacher, economics teacher, or some other teacher.

New assessments are expected for 2011-2012. They will probably still be CSAPs, just not as we currently know them. There is a push by CDE to make them more formative rather than summative, and there is some possibility that they might, at least in part, become computer-based to speed data collection, scoring, and data dissemination.

Previous high school CSAPs have specified only one of the three testing sessions where students could use an approved calculator. CDE now expects calculators and appropriate technology to be used for all high school content, as mastery of arithmetic should have been shown prior to high school. Teachers will no longer have to worry about teaching students a bit of math using their calculators only to find that it's assessed on a non-calculator portion of the CSAP.

Probability and statistics take a larger role in the new standards, which likely means they will be as big a part of the CSAP as algebra. The new probability and statistics content was designed using the GAISE report that I wrote about previously.

High school standards are no longer organized by grade. I know from personal experience it was difficult to deliver a geometry course to sophomores (nevermind the non-sophomores who might also be taking geometry) using a text that was primarily geometry, when I knew the 10th grade standards asked for mastery of topics spread across all six standards. For the revised standards, CDE decided to "bucket" the standards. Grade levels are no longer important at the high school level. When a student has practiced geometry content and is ready to show mastery, then they will take the geometry assessment. That might be 9th grade for some, 12th grade for others, and it will be up to each district/school to decide how to make their curriculum work. The Standards Review Committee was specifically told to complete the standards revision without regard to assessment, so there is still a great deal of work to do here before the structure and schedule of the new CSAPs is determined.

I wouldn't call the revisions radical, but Colorado districts and schools are going to do their homework and quickly modify their curriculum to meet the new standards. I think high school has the largest challenge because the assessment will probably look so different, but assessing by content, not grade level, was a long-overdue change. As someone who once taught a single section of Algebra 1 with every grade represented from 7 through 12, "bucketing" the high school standards is a most welcome change!

CCTM 2009: Guidelines for Assessment and Instruction in Statistics Education: Progress from K to 12

Presenter: Jerry Moreno, John Carroll University

I attended this session because I'm very interested in the growth of probability and statistics education in K-12 mathematics. I realize now that it was a neglected part of my own education, but as a teacher I find statistics interesting, relevant, and powerful. I also realize, however, that placing greater demands on students to learn probability and statistics means new curricula needs to be developed and other areas of math might be compromised.

Moreno made his feelings towards high school math sequences clear up front: integrated math sequences are the only way to go, and he's helping push the effort in Ohio to rid schools of the traditional Algebra 1-Geometry-Algebra 2 sequence. (I believe he said Ohio was trying to standardize class names such as "Math Reasoning 1, 2, and 3" in their place.) Moreno sees probability and statistics as the single largest piece of a puzzle that connects mathematics, science, and social studies, and much could be gained by increasing the use of data analysis in all three subjects.

Moreno's presentation summarized many of the efforts put into the GAISE (Guidelines for Assessment and Instruction in Statistics Education) report, an effort of the American Statistical Association to guide implementation of the NCTM Data Analysis and Probability standard. The ASA has produced a collection of student investigations called "Making Sense of Statistical Studies" for upper middle school and high school students. (They're working on sets of investigations for lower grades, building off work found in ASA's Statistics Teacher Network (STN) and other sources. Collectively the projects are known as GAP, or GAISE Activity Project.) During the session we discussed/experienced several quality investigations, dealing with varied topics such as the fairness of pennies, Mentos and Diet Coke, growing dahlias, and death certificate analysis.

There is clearly no lack of quality content available to teach probability and statistics to all levels. It remains to be seen, however, if schools are willing to break with traditions and reallocate their precious time to include more probability and statistics. I'm still wondering where the textbook publishers stand on this, particularly those who sell the traditional Algebra 1-Geometry-Algebra 2 series. Could we ever see a Algebra 1-Prob/Stats-Algebra 2 series, where geometry is reduced to supplementary materials, as many teachers have to do now for probability and statistics? Or will the massive educational inertia be too resistant to such a change?

CSAP Item Treemaps

Colorado's NCLB standardized testing program, the Colorado Student Assessment Program (CSAP) is administered to all Colorado public school students in grades 3-10 during late March or early May. Math is given to each grade as a series of three, one-hour testing sessions.

In my experience, the CSAP is very comprehensive, but getting details about the content of the test is a bit tricky. Colorado's standards are written for K-12; its benchmarks detail the standards further for spans of grade levels (K-4, 5-8, and 9-12). These are in no way detailed enough to design a curriculum, so the next level of detail is the assessment framework - the (generally) specific list of grade-specific topics from which the CSAP is designed. Analysis at this level might bring accusations of "teaching to the test," but CDE doesn't provide schools with much further guidance.

The assessment frameworks are a prime example of standards being a mile wide and an inch deep. They suggest quality content, for sure, but such a massive amount of it that most teachers I've seen become a bit bewildered when trying to comprehend it all. With a finite amount of time to teach this content, should a teacher prioritize? If so, how?

I've known and seen teachers who take what I consider to be an unethical shortcut: each spring they riffle through the test books and note the questions in the test. Those same teachers argue that because they aren't copying the problems or planning to use them verbatim, it isn't unethical. I argue that if CDE wanted teachers to see they test, they'd release the tests each year. CDE hasn't even released a single math test item since 2005.

CDE does release, however, item maps that indicates details about every single test item, including assessment framework objective and difficulty. The data does not come without a disclaimer, however. From the CDE website:

Purpose of Item Maps
The item maps contain information that may be of some assistance examining a school or [sic] districts adopted curricular alignment to the state standards. They are not an instructional tool, and cannot be used to develop curriculum.

Please refer to the Standards for Educational and Psychological testing relative to the ethical and appropriate use of assessment data, including item maps.


Item maps must not be used to create yearly instructional targets. Please keep in mind that objectives are assessed on a cyclical basis, and item focused instruction based on item map information is not only ineffective, it is an unethical use of the information provided.

While I admire CDE for wanting their data to be used in an ethical manner, the offering of item maps under these conditions comes across as a bit of a tease. CDE has drawn a very fine line when they say the data can be used to evaluate the alignment of curriculum to the standards, but cannot be used to actually develop any curriculum. Does this mean as long as your modifying a curriculum and not building one from scratch, you're okay? Surely that's not what CDE meant, but it does come across that way.

Back to the prioritizing problem: how do teachers prioritize content in their curriculum to meet the demands of the standards in a finite school year? In my case, my school year has been particularly finite - I first taught in a block-scheduled, single-semester (of an average of 85 days) system, then moved to a 4-day week school with a 144-day school year. Sure, our periods might have been a few minutes longer than a traditional 180-day school, but in math that doesn't usually translate into more lessons being taught. We were 36 days shorter than 180-day schools, the equivalent of one whole quarter for us. That's 468 days total for K-12, or an extra 3.25 144-day school years. Prioritization is necessary for performance.

After several years of struggle, I turned to the item maps in the hopes I could better align (not just what was taught, but for how long) my curriculum to the standards. As a mathematician, I knew the potential pitfals of focusing too narrowly on a single year of testing, but when I started there was five years' worth of high school item map data. I combined, averaged, and summarized the best I could, and learned more about the structure and progression of the standards than I ever thought I would. (I'm embarrassed now to admit how little I knew when I started teaching.) My final product, a new set of spreadsheets with relative rankings of the objectives within each standard, was complete. I showed it to a few other teachers, but they didn't seem to find them as useful as I had. I had learned through the process of building them, not from the final product, and other teachers seemed to get the same overwhelming feeling at looking at my spreadsheets as they would have gotten from the original item maps.

In June of 2009 I decided to tackle the problem again, this time with a goal of visually representing the data. I originally wanted a circle chart where you could drill-down into each slice to reveal details at the standard, benchmark, and assessment framework levels, but the coding proved to complicated. My searching for a solution led me to treemaps and javascript that I could easily work with. Here, visually, are the treemaps for each grade level:

Objectives are easily prioritized by shape and color, and the aggregation of many year's worth of data helps avoid overlooking topics that might not be addressed on any one particular year's test. Is this "teaching to the test?" Perhaps. That's a common battle cry against standardized education. But if there are standards to meet, and a generally well-designed, comprehensive test that measures students' performance on those standards, and item map data that can help us summarize the content of that test so it can be better prioritized in a finite school year, why not use it? If you have a better set of data or a better argument against using it, I'd love to hear it.

The Myth of the Slipping Math Student

I've been teaching in Colorado for six years, and there's always been a troubling pattern in our state standardized math scores. As students progress from 3rd to 10th grade, the percentage that score proficient and advanced declines dramatically. Here are the percentages of students scoring proficient and advanced by grade level, averaged over all the years the test has been given (typically 2002-2008):

Avg. % P+A

The easiest explanation (and the one I've tended to believe) is that students' abilities are, in fact, slipping as they got older. That would be a good assumption if the test at each grade level was equally difficult. But what if the test questions were, on average (and adjusted for grade level), more difficult as students got older? Is it fair to assume a test with increasingly difficult questions would result in lower scores, even with sophisticated score scaling systems that take question difficulty into account?

Fortunately, the state releases "item maps" that describe the difficulty of each item on every test. Using 4 points for an advanced item, 3 points for a proficient item, 2 points for a partially proficient item, and 1 point for an unsatisfactory item, we can come up with an average difficulty for the CSAP at each grade level. Let's add that column to our table:

Avg. Difficulty
Avg. % P+A

This begs for regression analysis. How strong is the correlation between the difficulty of the questions and the scores?

The correlation is surprisingly strong, and the coefficient of determination (R squared) is 0.88, meaning that the average item difficulty is statistically responsible for 88% of the variance in the test scores. 88%? That's big. Statistics rarely tell the whole story, but 88% raises serious doubts that it's just a matter of slipping math students. Why wouldn't the state want to maintain a steady average difficulty year-to-year? Wouldn't that make year-to-year performance comparisons more reliable?

Note: This was originally posted at http://johnson.downclimb.com/2009/06/myth-of-slipping-math-student.html.

First Day of School

It seemed fitting to me that the first post on MathEd.net should be like the first day of school.  So please sit in your seats quietly and patiently as I go on for the entire class period telling you what you aren't allowed to do here.

Just kidding.

But isn't that a traditional expectation for the first day?  I tried it that way (several times, I'll admit) and it had to rank as some of the worst teaching I've ever subjected my students to.  I read somewhere once that if you spend the entire first day of school addressing the rules, then you'd better be prepared to deal with them the rest of the year.  I agree.  Instead, I made sure students were involved in a group problem-solving activity the moment they walked in the door.  Rules weren't discussed, but the expectations for my students were clear.

The only rules I had posted in my classroom were my "Math Rules."  You'll have to pardon some of the sarcasm, but I've had success using this kind of humor with students.  (Besides, it's just who I am.)

Math Rules

  1. Don't disrespect zero and one. They're your friends. You wouldn't "cancel" a friend, would you?
  2. Exact answers are usually better than approximate ones.  Why would you want to do more work to change an exact fraction into an approximate decimal?  Yeah, thought so. If you must "decimalize," two decimal places is usually enough.
  3. Of course you have to show your work. Duh! I can't believe you'd even ask such a silly question. The same goes for reducing fractions.
  4. Generally, if a complete sentence was used to ask for an answer, respond with a complete sentence. It’s good for you, like vegetables.
  5. Don't "plus" things together. Don't "times" things together. Know when to say "add" and "multiply" so you won't sound like a dork.

This list helped accomplish three objectives. First, build rapport with students through humor. Second, establish that how we communicate about math is important. Third, help students become better mathematicians by setting expectations for their work. I think the rules worked well, especially with #5 - students started to look for opportunities to call each other "dorks" and incorrect use of phrases like "I timesed the numbers together" decreased substantially.