Tag Archives: SBG

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!

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Filed under algebra 2, calculus, geometry, physics, reflection, teaching philosophy, Uncategorized

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.

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Rethinking the headache of reassessments with Python


One of the challenges I’ve faced in doing reassessments since starting Standards Based Grading (SBG) is dealing with the mechanics of delivering those reassessments. Though others have come up with brilliant ways of making these happen, the design problem I see is this:

  • The printer is a walk down the hall from my classroom, requires an ID swipe, and possibly the use of a paper cutter (in the case of multiple students being assessed).
  • We are a 1:1 laptop school. Students also tend to have mobile devices on them most of the time.
  • I want to deliver reassessments quickly so I can grade them and get them back to students immediately. Minutes later is good, same day is not great, and next day is pointless.
  • The time required to generate a reassessment is non-zero, so there needs to be a way to scale for times when many students want to reassess at the same time. The end of the semester is quickly approaching, and I want things to run much more smoothly this semester in comparison to last.

I experimented last fall with having students run problem generators on their computers for this purpose, but there was still too much friction in the system. Students forgot how to run a Python script, got errors when they entered their answers incorrectly, and had scripts with varying levels of errors in them (and their problems) depending on when they downloaded their file. I’ve moved to a web form (thanks Kelly!) for requesting reassessments the day before, which helps me plan ahead a bit, but I still find it takes more time than I think it should to put these together.

With my recent foray into web applications through the Bottle Python framework, I’ve finally been able to piece together a way to make this happen. Here’s the basic outline for how I think I see this coming together – I’m putting it in writing to help make it happen.

  • Phase 1 – Looking Good: Generate cleanly formatted web pages using a single page template for each quiz. Each page should be printable (if needed) and should allow for questions that either have images or are pure text. A function should connect a list of questions, standards, and answers to a dynamic URL. To ease grading, there should be a teacher mode that prints the answers on the page.
  • Phase 2 – Database-Mania: Creation of multiple databases for both users and questions. This will enable each course to have its own database of questions to be used, sorted by standard or tag. A user can log in and the quiz page for a particular day will automatically appear – no emailing links or PDFs, or picking up prints from the copier will be necessary. Instead of connecting to a list of questions (as in phase 1) the program will instead request that list of question numbers from a database, and then generate the pages for students to use.
  • Phase 3 – Randomization: This is the piece I figured out last fall, and it has a couple components. The first is my desire to want to pick the standard a student will be quizzed on, and then have the program choose a question (or questions) from a pool related to that particular standard. This makes reassessments all look different for different students. On top of this, I want some questions themselves to have randomized values so students can’t say ‘Oh, I know this one – the answer’s 3/5’. They won’t all be this way, and my experience doing this last fall helped me figure out which problems work best for this. With this, I would also have instant access to the answers with my special teacher mode.
  • Phase 4 – Sharing: Not sure when/if this will happen, but I want a student to be able to take a screenshot of their work for a particular problem, upload it, and start a conversation about it with me or other students through a URL. This will also require a new database that links users, questions, and their work to each other. Capturing the conversation around the content is the key here – not a computerized checker that assigns a numerical score to the student by measuring % wrong, numbers of standards completed, etc.

The bottom line is that I want to get to the conversation part of reassessment more quickly. I preach to my students time and time again that making mistakes and getting effective feedback is how you learn almost anything most efficiently. I can have a computer grade student work, but as others have repeatedly pointed out, work that can be graded by a computer is at the lower level of the continuum of understanding. I want to get past the right/wrong response (which is often all students care about) and get to the conversation that can happen along the way toward learning something new.

Today I tried my prototype of Phase 1 with students in my Geometry class. The pages all looked like this:

Image

I had a number of students out for the AP Mandarin exam, so I had plenty of time to have conversations around the students that were there about their answers. It wasn’t the standard process of taking quiz papers from students, grading them on the spot, and then scrambling to get around to have conversations over the paper they had just written on. Instead I sat with each student and I had them show me what they did to get their answers. If they were correct, I sometimes chose to talk to them about it anyway, because I wanted to see how they did it. If they had a question wrong, it was easy to immediately talk to them about what they didn’t understand.

Though this wasn’t my goal at the beginning of the year, I’ve found that my technological and programming obsessions this year have focused on minimizing the paperwork side of this job and maximizing opportunities for students to get feedback on their work. I used to have students go up to the board and write out their work. Now I snap pictures on my phone and beam them to the projector through an Apple TV. I used to ask questions of the entire class on paper as an exit ticker, collect them, grade them, and give them back the next class. I’m now finding ways to do this all electronically, almost instantly, and without requiring students to log in to a third party website or use an arbitrary piece of hardware.

The central philosophy of computational thinking is the effort to utilize the strengths of  computers to organize, iterate, and use patterns to solve problems.  The more I push myself to identify my own weaknesses and inefficiencies, the more I am seeing how technology can make up for those negatives and help me focus on what I do best.

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Why SBG is blowing my mind right now.


I am buzzing right now about my decision to move to Standards Based Grading for this year. The first unit of Calculus was spent doing a quick review of linear functions and characteristics of other functions, and then explored the ideas of limits, instantaneous rate of change, and the area under curves – some of the big ideas in Calculus. One of my standards reads “I can find the limit of a function in indeterminate form at a point using graphical or numerical methods.”

A student had been marked proficient on BlueHarvest on four out of the five, but the limit one held her back. After some conversations in class and a couple assessments on the idea, she still hadn’t really shown that she understood the process of figuring out a limit this way. She had shown that she understood that the function was undefined on the quiz, but wasn’t sure how to go about finding the value.

We have since moved on in class to evaluating limits algebraically using limit rules, and something must have clicked. This is what she sent me this morning:

Getting things like this that have a clear explanation of ideas (on top of production value) is amazing – it’s the students choosing a way to demonstrate that they understand something! I love it – I have given students opportunities to show me that they understand things in the past through quiz retakes and one-on-one interviews about concepts, but it never quite took off until this year when their grade is actually assessed through standards, not Quiz 1, Exam 1.

I also asked a student about their proficiency on this standard:

I can determine the perimeter and area of complex figures made up of rectangles/ triangles/ circles/ and sections of circles.

I received this:
…followed by an explanation of how to find the area of the figure. Where did she get this problem? She made it up.

I am in the process right now of grading unit exams that students took earlier in the week, and found that the philosophy of these exams under SBG has changed substantially. I no longer have to worry about putting on a problem that is difficult and penalizing students for not making progress on it – as long as the problem assesses the standards in some way, any other work or insight I get into their understanding in what they try is a bonus. I don’t have to worry about partial credit – I can give students feedback in words and comments, not points.

One last anecdote – a student had pretty much shown me she was proficient on all of the Algebra 2 standards, and we had a pretty extensive conversation through BlueHarvest discussing the details and her demonstrating her algebraic skills. I was waiting until the exam to mark her proficient since I wanted to see how student performance on the exam was different from performance beforehand. I called time on the exam, and she started tearing up.

I told her this exam wasn’t worth the tears – she wanted to do well, and was worried that she hadn’t shown what she was capable of doing. I told her this was just another opportunity to show me that she was proficient – a longer opportunity than others – but another one nonetheless. If she messed up a concept on the test from stress, she could demonstrate it again later. She calmed down and left with a smile on her face.

Oh, and I should add that her test is looking fantastic.

I still have students that are struggling. I still have students that haven’t gone above and beyond to demonstrate proficiency, and that I have to bug in order to figure out what they know. The fact that SBG has allowed some students to really shine and use their talents, relaxed others in the face of assessment anxiety, and has kept other things constant, convinces me that this is a really good thing, well worth the investment of time. I know I’m just preaching to the SBG crowd as I say this, but it feels good to see the payback coming so quickly after the beginning of the year.

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Filed under algebra 2, calculus, reflection, teaching philosophy, teaching stories

Standards Based Grading – All in, for the new year


I’ve written previously about wanting to be part of the Standards Based Grading crowd. My quiz policy was based in the idea – my quizzes cover skills only and in isolation, the idea being that if students could show proficiency on the quizzes, then I would know for sure that they had really developed those skills. If they had demonstrated proficiency, but then failed on tests to perform, it was an indication that the problem was seeing all the skills in one place. This is the “I get it in class, but on tests I mess it up” mantra that I’ve heard ever since I first started teaching. My belief has always been that the first clause of that sentence is never as true as the student thinks it is. The quiz grades have typically shown that to be the case.

The thing I haven’t been able to get at is why I can’t get my students to retake quizzes as I thought it compelled them to do. I told them they can get 100%. I reminded them that they just needed to look at each quiz, recognize what they got wrong, and work with me on those specific skills to improve. Then, when they were ready, they could retake and get a better score. Sometimes they do it, but they are always missing either one of those three things. They would retake without looking at the quiz. They would take it knowing what they got wrong, but never asked me to go over the things they didn’t get. There were exceptions, but curiously not enough to impress me.

After really committing to reshaping the quiz grade as a real SBG grade for a unit last year, I saw the differences pretty clearly in how the students went about this aspect of their grade. The standards I expected students to demonstrate were clearly listed in the grade book (fine, Powerschool). The students knew what they needed to work on, and were directly linked to examples and short videos I had created to help them with those specific skills. Class time was spent working around developing those skills, along with some bigger picture ideas to explore separately from the routine skills the standards were centered around for the unit, which was on exponential and logarithmic functions. I was impressed in this short time with how changing this small (15%) portion of the grade changed the overall attitude my students had while they were working with me. It was one step closer to the Montessori style classroom I have always wanted to have while working within the structure of a more traditional program – students walk in knowing what they need to work on, and they get to work. My role becomes more to push them in the way I think they can and need to be pushed. Some need to work on skills, others need to attack context problems and the challenging ‘why is this so’ threads that are usually all teacher driven, but don’t need to be in many cases.

I did some thinking over the last couple of weeks on how I wanted to do things differently, so I wrote up a new grading policy and posted it online. I had renamed my quiz grade to be ‘Learning Standards’, bumped up the percentage by 10% (to 25%), and reduced the homework and classwork components to 5% each, with a portfolio at 10%, and tests to 55%.  In sharing my new grading policy with people through Twitter, there were some key comments that really guided my thinking.

Kelly O’Shea pointed out the fact that even with the change, the standards were not a huge part of the grade. Even by cutting classwork and homework into the standards, it still wasn’t good enough:

A few other people made similar suggestions. John Burk probably put the final nail in the SBG-lite version I thought was safe with this comment:

One problem for getting buy in on SBG is that if it isn’t a big part of the grade, and there are still so many non-sbg things, they might not really understand the rationale for SBG.

If I really believe in the power for Standards Based Grading to transform how learning happens in my classroom, I need to demonstrate its importance and commit to it.

The final result? My grades for Algebra 2/Advanced Algebra, Geometry, Calculus 12, and Physics are going to be 90% Learning Standards, 10% portfolio. I am going to give unit tests, but they are opportunities to demonstrate proficiency on the learning standards. In the case of my AP Calculus students, the grades are still 60% unit tests, 30% standards, and 10% portfolio, primarily because I still will be giving tests that are similar to the AP exam with multiple choice, and free response sections. I also had my first class last year with 100% fives, and am admittedly a bit nervous tweaking what worked last year. That said, I am accepting that this, too, could become a thing of the past.

I am a bit nervous, but that’s mostly because change isn’t always easy. From a teaching perspective, the idea feels right, but it’s not what I’m used to doing. The students sounded pretty cool with it on the first days of class when I introduced the idea though, and that is a major positive. I’ll keep writing as things proceed and my implementation develops – it feels great to know I’m not alone.

I really appreciate all of the kind words and honest feedback from the people that challenged me to think this through and go all in. If I can do nothing else, I’ll pay that advice forward. Cool?

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End of year reflections – SBAR analysis


I wrapped up grading final exams today. Feels great, but it was also difficult seeing students coming in to get their final checklists signed before they either graduate or move on to a different school. Lots of bittersweet moments in the classroom today.

I decided after trying my standards based grading (SBG) experiment that I wanted to compare different students’ overall performances among the grading categories to their quiz percentages. In a previous post, I wrote about my experimentation doing my quiz assessments on very specific skill standards that the students were given. As I plan to change my grading to be more SBG based for the fall, I figured it would be good to have some comparison data to be able to argue the many reasons why this is a good idea.

In my geometry and algebra two classes, there are 28 total students. I removed two students from the data set that came in the last month of school, and one outlier in terms of overall performance.

The table below shows the names of each grading category, as well as the overall grade weight for the category used in calculating the grade. The numbers are the correlation coefficient in the data between the variable listed in the row and column. For example, the 0.47 is the correlation between the HW data and the Quizzes/Standards data for each student.


Homework (8%) Quizzes/Standards (12%) Test Average (48%) Semester Exam (20%) Final Grade (100%)
HW 1 0.47 0.41 0.36 0.48
Quiz/Stndrds
1 0.89 0.91 0.95
Tests

1 0.88 0.97
Sem. Exam


1 0.95
Final Grade



1

Some notes before we dive in:

  •  The percentages do not add up to 100% because I am leaving out the portfolio grade (8%) and classwork grades (4%) which are not performance based.
  • Homework is graded on turning it in and showing work for answers. I collect and look at homework as a regular way to assess how well they are learning the material.
  • The empty cells are to save ink; they are just the mirror image of the results in the upper half of the matrix since correlation is not order dependent.
  • I know I only have 25 samples – certainly not ready for a research publication.

So what does this mean? I don’t know that this is very surprising.

  1. Students doing HW is not what makes learning happen. I’ve always said that and this continues to support that hypothesis. It can help, it is evidence that students are doing something with the material between classes, but simply completing it is not enough. I’m fine with this. I get enough information from looking at the homework to create activities that flesh out misunderstanding the next time we meet. The unique thing about homework is that it is often the first time students look at the material on their own rather than with their peers in class.
  2. My tests include some direct assessments of skills, but also include a lot of new applications of concepts and questions requiring students to explain or show things to be true. It’s very gratifying to see such a strong connection between the quiz scores and the test scores.
  3. I always wonder about the students that say “I get it in class, but then on the tests I freeze up.” If there’s any major lesson that SBG has confirmed for me, it’s that student self-awareness of proficiency is generally not great without some form of external feedback. If this were the case, there would be more data with high quiz scores and low exam scores. That isn’t the case here. My students need real and correct feedback on how they are doing, and the skills quizzes are a formalized way to do this.
  4. I find it really interesting how close the quiz average and the semester exam percentages are. The semester exam was cumulative and covered a lot of ground, but it didn’t necessarily hit every single skill that was tested on quizzes. There were also not quizzes for every single skill, though I tried to hit a number of key ones.

This leads me to believe that it is possible to have several key standards to focus on for SBG purposes, and also to dedicate time to work on other concepts during class time through project based learning, explorations, or independent work. It’s feasible to assess these other concepts as mathematical process standards that are assessed throughout the semester. It strikes a good balance between developing skills according to curriculum but not making classes a repetitive process of students absorbing procedures for different types of problems. I want to have both. My flipping experiments have worked well to approaching that ideal, but I’m not quite there yet.

I’ll have more to say about the details of what I will change as I think about it during the summer. I think a combination of using BlueHarvest for feedback, extending SBG to my Calculus and Physics classes, and less emphasis on grading and collecting homework will be part of it. Stay tuned.

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Results of a unit long experiment in SBG and flipping.


I’ve been a believer in the concept of standards based instruction for a while. The idea made a lot of sense when I first learned about the idea when Grant Wiggins visited my school in the Bronx a few years ago to present on Understanding by Design. Dan Meyer explored the idea quite a bit using his term of the concept checklist. Shawn Cornally talks on his blog about really pushing the idea to give students the freedom to demonstrate their learning in a way they choose, though he ultimately retains judgment power on whether they have or not. Countless others have been really generous in sharing their standards and their ideas for making standards work for their students. Take a look at my blogroll for more people to read about. For those unaware, here’s the basic idea: Look at the entire unit and identify the specific skills or you want your students to have. Plan your unit to help them develop those skills. Assess and give students feedback on those skills as often as possible until they get it. In standards based grading (SBG), reporting a grade (as most of us are required to do) as a fraction of standards completed or acquired becomes a direct reflection of how much students have learned. Compare this to the more traditional version of grading that consists of an average of various ‘snapshots’ on assignments, on which grades might be as much a reflection of effort or completion as of actual learning. If learning is to be the focus of what we do in the classroom, then SBG is a natural way of connecting that learning to the grades and feedback we give to students. My model for several years now has been, well,  SBG lite. Quizzes are 15% of the total grade and test only a couple skills at a time. Students can retake quizzes as many times as they want to show that they have the skills in isolation. On tests, (60% of the total grade) students can show that they can correctly apply the set of all of their acquired skills on exercises (questions they have seen before) as well as problems (new questions that test conceptual understanding). As much as I tell students they can all have a grade of 100% for quizzes and remind those that don’t to retake, it doesn’t happen. I’ll get a retake here or there. I am still reporting quiz grades as an average of a pool of “points” though, and this might leave enough haziness in the meaning of the grade for a student to be OK with a 60%. For this unit in Geometry and Algebra 2, I have specifically made the quiz grade a set of standards to be met. The point total is roughly the same as in previous units. It is a binary system – students either have the standard (3/3) or they don’t (0/3), and they need to assess each standard at least twice to convince me they have it. I really like Blue Harvest, but my students didn’t respond so well to having twowhole websites to use to check progress. While a truly scientific study would have changed only one variable at a time, I also found that structuring the skill standards this way required me to change the way class itself was structured. This became an experiment not only in reporting grades, but in giving my students the power to work on things in their own way. This also freed me up to spend my time in class assessing, giving feedback, and assessing again. More on this ahead. The details:

Geometry

I started the unit by defining the seven skills I wanted the students to have by the end on this page. The unit was on transformational geometry, so a lot of the skills were pretty straight forward applications of different types of transformations to points, line segments, and polygons. I had digital copies of all of the materials I put together last year for this unit, so I was able to post all of that material on the wiki for students to work through on their own. I adjusted these materials as we moved through the unit and as I saw there were holes in their understanding. I was also able to make some videos using Jing and Geogebra to explain some concepts related to using vocabulary and symmetry, and these seemed to help some students that needed a bit of direct instruction in addition to what I provided to them one on one. I also tried another experiment – programming assignments related to applying transformations to various points. I said completing these assignments and chatting with me about them would qualify them for proficiency on a given standard. Assigning homework was simple: Choose a standard or two, and do some of the suggested problems related to those standards. Be prepared to show me your evidence of study when you come into class. Students that said ‘I read my notes’ or ‘I looked it over’ were heckled privately – the emphasis was on actively working to understand concepts. Some students did flail a bit with the new freedom, so I made suggestions for which standards students should spend a particular day working on, and this helped these students to focus. I threw together some concept quizzes for the standards covered by the previous classes, and students could choose to work on those question types they felt they had mastered. Some handed the quiz right back knowing they weren’t ready. I was really pleased with the level of awareness they quickly developed around what they did and didn’t understand. I quickly ran into the logistical nightmare of managing the paperwork and recording assessment results. Powerschool Blue Harvest, whatever – this was the most challenging aspect of doing things this way. I often found myself bogged down during the class period recording these things, which got in the way of spending quality face time with students around their understanding. Part of this was that I was recording progress for each standard, whether good or bad, in the comment field for each student. “Understands basic idea of translation, but is confusing the image and pre-image” is the sort of comment I started writing in the beginning. While this was nice, and I think could have led to students reading the comments and getting ideas for what they needed to work on, it was a bit redundant since I was having actual conversations with students about these facts. Here is where Blue Harvest shines – I can easily send students a quick message explaining (and showing) what they need to work on. Even more powerful would be recording the conversation when I actually talk to the student, but that would be more practical with an iPad/cell phone app to avoid lugging my computer from desk to desk. Still, I wanted the feedback to be immediate and be recorded, so I knew I had to change my approach. The compromise was to only record positive progress. If a student’s quiz showed no progress, it didn’t get a comment in Powerschool. If they showed progress, but needed to fix a small detail in their understanding, they might get a comment. If they clearly got it, they got a comment saying that they aced it. Two or more positive comments (and my independent review) led to a 3/3 for each standard. The other promise I made was that if they clearly demonstrated proficiency on the exam (which had non-standard questions and some things they needed to explain) I would give them credit for the standard. The other difficult issue was creating a bank of reassessment questions. My system of making a quiz on the spot and handing it out to individual students was too time consuming. I created an app(using my new Udacity knowledge) to try to do this, the centerpiece being a randomized set of questions that emphasized knowing how to figure out the answers rather than students potentially sharing all the answers. They quickly found all the bugs in my system, and showed that it is far from ready for being an actual useful tool for this purpose. I appreciated their humor and patience in being guinea pigs for an idea. As you might notice from the image above, there is a pretty strong relationship between the standards mastered and the exam scores. Most student exam scores were either the same or better following this system in comparison to previous exams. The most important metric is the fact that most students weren’t hurt by going to this more student-centered model. Some student took more notes while working to understand the material than they have all year. Other students spoke more to their classmates and both gave and received more help in comparison to when I was at the front of the room asking questions and doing mini-lessons. While there was a lot of staring at screens during this unit, there was also a lot of really great discussion. I would have focused conversations with every single student three to four times a class, and they were directly connected to the level of understanding they had developed. Some needed direct application questions. Others could handle deeper synthesis and ‘why is this true’ questions about more abstract concepts. It felt really great doing things this way. I have always insisted on crafting one good solid presentation to give the class – the perfect lesson – with good questions posed to the class and discussions inevitably resulting from them. I have to admit that having several smaller, unplanned, but ‘messier’ conversations to guide student learning have nurtured this group to be more independent and self driven than I expected before we started.

Algebra 2

The unit focused on the students’ first exposure to logarithmic and exponential functions. The situation in Algebra 2 was very similar to Geometry, with one key difference. The main difference of this class compared to Geometry is that almost all of the direct instruction was outsourced to video. I decided to follow the Udacity approach of several small videos (<3 min), because that meant there was opportunity (and the expectation) that only two minutes would go by before students would be expected to do something. I like this much better because it fit my own preferences in learning material with the Udacity courses. I had 2 minutes to watch a video about hash functions in Python while brushing my teeth – my students should have that ability too. I wasn’t going for the traditional flipped class model here. My motivation was less about requiring students to watch videos for homework, and more about students choosing how they wanted to go through the material. Some students wanted me to do a standard lesson, so I did a quick demonstration of problems for these students. Others were perfectly content (and successful) watching the video in class and then working on problems. Some really great consequences of doing things this way:

  • Students who said they watched all my videos and ‘got it’ after three, two minute videos, had plenty of time in the period to prove it to me. Usually they didn’t.. This led to some great conversations about active learning. Can you predict the next step in the video when you try solving the problem on your own? What? You didn’t try solving it on your own? <SMIRK>  The other nice thing about this is that it’s a reinvestment of two minutes suggesting that they try again with the video, rather than a ten or fifteen minute lesson from Khan Academy.
  • I’ve never heard such spirited conversation between students about logarithms before. The process of learning each skill became a social event – they each watched the video together, rewound or paused as needed, and then got into arguments while trying to solve similar problems from the day’s handout. Often this would get in the way during teacher-centered lessons, and might be classified incorrectly as ‘disruption’ rather than the productive refining and conveyance of ideas that should be expected as part of real learning.
  • Having clear standards for what the students needed to be able to do, and making clear what tools were available to help them learn those specific standards, led to a flurry of students demanding to show me that they were proficient. That was pretty cool, and is what I was trying to do with my quiz system for years, but failed because there was just too much in the way.
  • Class time became split between working on the day’s standards, and then stopping at an arbitrary time to then look at other cool math concepts. We played around with some Python simulations in the beginning of the unit, looked at exponential models, and had other time to just play with some cool problems and ideas so that the students might someday see that thinking mathematically is not just followinga list of procedures, it’s a way of seeing the world.

I initially did things this way because a student needed to go back to the US to take care of visa issues, and I wanted to make sure the student didn’t fall behind. I also hate saying ‘work on these sections of the textbook’ because textbooks are heavy, and usually blow it pretty big. I’m pretty glad I took this opportunity to give it a try. I haven’t finished grading their unit exams (mostly because they took it today) but I will update with how they do if it is surprising.


Warning: some philosophizing ahead. Don’t say I didn’t warn you. I like experimenting with the way my classroom is structured. I especially like the standards based philosophy because it is the closest I’ve been able to get to recreating my Montessori classroom growing up in a more traditional school. I was given guidelines for what I was supposed to learn, plenty of materials to use, and a supportive guide on the side to help me when I got stuck. I have seen a lot of this process happening with my own students – getting stuck on concepts, and then getting unstuck through conversation with classmates and with me. The best part for me has been seeing my students realize that they can do this on their own, that they don’t always need me to tell them exactly what to do at all times. If they don’t understand an idea, they are learning where to look, and it’s not always at me. I get to push them to be better at what they already know how to do rather than being the source of what they know. It’s the state I’ve been striving to reach as a teacher all along, and though I am not there yet, I am closer than I’ve ever been before. It’s a cliche in the teaching world that a teacher has done his or her job when the students don’t need you to help them learn anymore. This is a start, but it also is a closed-minded view of teaching as mere conveyance of knowledge. I am still just teaching students to learn different procedures and concepts. The next step is to not only show students they can learn mathematical concepts, but that they can also make the big picture connections and observe patterns for themselves. I think both sides are important. If students see my classroom as a lab in which to explore and learn interesting ideas, and my presence and experience as a guide to the tools they need to explore those ideas, then my classroom is working as designed. The first step for me was believing the students ultimately wantneed to know how to learn on their own. Getting frustrated that students won’t answer a question posed to the entire class, but then will gladly help each other and have genuine conversations when that question comes naturally from the material. All the content I teach is out there on the internet, ready to be found/read/watched as needed. There’s a lot of stuff out there, but students need to learn how to make sense of what they find. This comes from being forced to confront the messiness head on, to admit that there is a non-linear path to knowledge and understanding. School teaches students that there is a prescribed order to this content, and that learning needs to happen within its walls to be ‘qualified’ learning. The social aspect of learning is the truly unique part of the structure of school as it currently exists. It is the part that we need to really work to maintain as content becomes digital and schools get more wired and connected. We need to give students a chance to learn things on their own in an environment where they feel safe to iterate until they understand. That requires us as teachers to try new things and experiment. It won’t go well the first time. I’ve admitted this to my students repeatedly throughout the past weeks of trying these things with my classes, and they (being teenagers) are generous with honest criticism about whether something is working or not. They get why I made these changes. By showing that iteration, reflection, and hard work are part of our own process of being successful, they just might believe us when we tell them it should be part of theirs.

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Filed under algebra 2, geometry, reflection, teaching philosophy, teaching stories

Using #Geogebra to Predict and then Verify


Last year’s class introducing logarithmic and exponential differentiation was a bust. I tried to include it as an application of implicit differentiation, but I knew afterwards then, and still believe now that doing so was an incredibly horrible idea. There’s no way students are going to ‘see’ an application of an abstract concept like implicit differentiation better…by using it in another abstract concept. I’ve accepted that, and vowed this year to do a much better job.

I also had a shocking moment yesterday when a Calculus student came to me after school and asked me ‘what is the derivative?’ We had started the unit with a conceptual development of the derivative using limits and average rate of change, and had since moved to applying differentiation rules, so we were deep in that process – power rule, quotient rule, product rule, chain rule…really the primary ‘rules’ section of any Calculus course. I was taken aback by the comment – had I really stopped emphasizing the definition of the derivative in our class activities? In a way, yes. We had been writing equations for tangent lines and graphing them, but we hadn’t seen the limit definition (which I’ve been impressed by students remembering) in a little while. This proved that not only did I need to do a better job with logs and exponential functions, but that a little conceptual basis in that process would be useful.

I always like using Geogebra as a tool to pre-load information I am about to give students – what is about to happen? What should my result look like when I do this on pencil and paper? The graphing capabilities make it really easy to do this and set this up – I created this file and made it look the way I wanted in a few minutes.

You can direct download the file here.

These were the instructions I gave students:

Sketch what you would expect the derivative of y = 2^x to look like. Then click the ‘Show Derivative Function’ to graph the actual derivative. How close were you?

How would you expect your sketch to change for the derivative of y = 3^x?

Graph and make a prediction of the graph of the derivative of y = 2^-x. Check and see how close you were using the Geogebra tool.

Can you adjust the slider value for a so that the derivative is the same as the function itself? Use the arrow keys to adjust the slider more precisely.

Go through this same process to sketch the derivative of y = ln(x) in a new Geogebra window. Create this by going to the ‘File’ menu and selecting ‘New Window’.

It was really great seeing students predicting what the derivative would be, and then using the applet to confirm what they thought. There were lots of good conversations about scale factors and reflections, and some of them pretty much nailed what the general forms were going to be. This made the algebraic derivation a piece of cake – they knew where it was headed.

I also sprung this on them:

I’ve been really getting into the idea of standard based grading, and have been doing a form of it through my quizzes for a while, but it is still a small component of the overall grade calculation. While their grades aren’t being calculated any differently at the moment, I shared that this list would make a really good tool as we prepare for the unit exam on derivatives next week, and most started going through on their own and deciding what they needed to work on.

I’m still getting caught up after a couple very busy weeks, but I really like how this group in Calculus has been developing and maturing as math students in only a couple months. Their questions are more directed: ‘I don’t understand this application of the chain rule’ compared to ‘I don’t get it’. Their written work is detailed and clear, making it easy to locate errors. As a group, they get along really well, and class periods are filled with moments of furious productivity and camaraderie as well as humor and smiles throughout.

It was raining hard all day. I watched some students walk into class, look outside at the afternoon sky, and sink into their chairs, clearly feeling a bit down. I told them it was perfect Calculus weather – why not sit inside and do some differentiation?

Probably not what they had in mind. By the end of class, everyone left the classroom looking much more positive than when they walked in, and at least feeling good about the work they had in front of them.

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